Camera Math of photography settings and EV

A few math details about EV, f/stop, shutter speed, and ISO

This page is Not likely of interest to those just wanting to take a picture. But if interested in computing with the camera exposure settings, here’s a few details which may be of interest. I try to keep the math isolated here, but the math relates to several calculators on the local site pages. Related pages are: EV and EV Chart on page evchart.html, and also the charts of camera nominal and precise setting values for page fstop2.html. Several of the calculators here use these same basics.

First, a Quick Summary of Exposure Computing

If these first basics are not clearly obvious, a longer version follows below.

The settings marked on cameras (shutter speed, f/stop, ISO) are Nominal values (approximate or rounded convention for human use), and are Not the exact Precise Goal Values that are used by the camera or in exposure calculations. For example, what we call 1/1000 second is actually binary 1/1024 second, which is unimportant in photography practice, but math needs the precise number. Actual camera design uses setting stops of exact powers of 2 for shutter speed values (1, 2, 4, 8, 16, 32, 64, 128, 256, etc), and powers of √2 for f/stop NUMBERS. ISO values are also powers of 2, but are modified to make ISO 100 become a full stop (and third stop numbers be relative to it). The precise goal of setting calculations are detailed below.
Differences of exposure in EV units are also the sequential powers of 2 (i.e., 1 EV difference is a factor of 2x). Log₂(X) (below) is simply the exponential power of 2 that equals X (and we measure exposure in powers of 2, called EV). Negative exponents compute fractions (less than 1), for example nominal shutter speed 1/30 second is the fifth sequential stop (counting from 1 second as stop number 0), so 2^-5 = 0.03125 = 1/32 second. This is the actual design goal. These sequential exponents of two are named to be Stop Number. Shutter speed 1 second and also fstop f/1 are stop number 0 because 2⁰ is 1 (any number to power of 0 is 1).
Some Quirks about f/stops:
Value of f/stop NUMBERS are necessarily squared first in exposure calculations (because f/stop NUMBERS are powers of √2 but exposure stops are powers of 2).
And EV ratios of f/stop numbers are reversed in the division for proper sign, and squared (because f/stop numbers increase exposure running in the opposite direction). If you get the wrong sign, you need to reverse the division.
Guide Number calculation is an exception, when f/stop is not squared, because Inverse Square Law of distance is also a √2 factor. Which is what makes Guide Numbers so handy, automatically compensating for the Inverse Square Law.
To compare f/11 to f/2.8 (which we know f/11 is -4 EV from f/2.8)
(f/stop ratios are reversed division due to f/stop numbers running backwards)
n = 2.828/11.314 = 0.25 (with precise values of f/stops)
EV = log₁₀(n²) / log₁₀(2) = -4 EV difference (log of fractions are negative)
Or n = 11.314/2.828 = 4 f/2.8 compared to f/11
EV = log₁₀(n²) / log₁₀(2) = +4 EV difference
But shutter speed and ISO are powers of 2, and run forward, so :
To compare 1/30 to 1/125 (which we know 1/30 is +4 EV from 1/125)
n = (1/32) / (1/128) = 128 / 32 = 4 (with precise values of shutter speeds)
EV = log₁₀(n) / Log₁₀(2) = 4 EV difference.
Shutter speed and ISO numbers and ratios are Not reversed or squared, only f/stops
Fractional stops (like third, half, or tenth stops) simply add the fraction to the exponent of 2 or √2. For example, 2/3 EV more than nominal f/5.6 (which is Stop Number 5) is √2^{(5 + 2/3)} = 1.41421^5.66667 = f/7.1272. It’s good for precision to use a couple of extra significant digits in the factors of exponents.
Logarithms in programming languages (such as Excel, Javascript, Python, C, or Fortran, etc.) use log_e instead of log₁₀. Which works fine, no issue at all, calculation result is the same, except mixing with assumed constants from the wrong base won’t match. For example, the common approximation log₁₀(2) = 0.30103 is valid in base 10, and just 0.3 is commonly used if greatest precision is not needed, but in programming languages, log_e(2) = 0.693147 (NOT 0.3).
Note that these programming languages will also expect trig angles to be radians instead of degrees. Radians = Degrees x Pi/180. Handheld calculators usually provide this choice, with degrees as default.

The formula in the second box at right is how light meter design computes exposure settings. N² / t reflects the camera settings, which appropriate exposure matches to the LS / K values. The meter knows ISO and meters the light and computes EV for the settings to match it. ISO matches the reflected scene luminance to the camera exposure settings of shutter speed and aperture.

For incident-light meters, N² / t = ES / C where
E is the illuminance (total luminous flux incident on a surface, per unit area)
C is the incident-light meter calibration constant (typically 340 with dome)

The Calibration Constant specification is shown in all Sekonic meter manuals as K = 12.5 for reflected meters, or as C = 340 for incident meters with domes. Nikon and Canon camera reflected meters are also said to use K = 12.5.

The calibration constants are a bit arbitrary, but the ISO specifications specifically say the calibration constant should be determined by analysis by large numbers of observers of a large number of photos of known exposure in various conditions. Meaning a visual consensus, since exact exposure is pretty hard to judge critically.

Other than knowing the idea, I imagine most of us have little need for the light meter formula, and the main math thing computing EV does need to know is:

EV = log₂

N²
t

which means 2^EV =

N²
t

EV 0 is f/1 and 1 second because 2⁰ = 1. The math just works out that way, since any number to power 0 is value 1, and log(1) of any base is 0. So any f/stop on the EV 0 row of the EV chart shows this relation (if using the precise values for f/stop and shutter speed shown below, instead of their nominal approximations).

N is f/stop Number and t is shutter speed duration Time. Note f/stop NUMBERS are squared in exposure calculations. The method to compute log₂ is defined below. Exponents and Logarithms are simply "reverse" operations, in the same way multiply and divide or add and subtract are reverse operations ("undoing" each other, so to speak).

Example: If f/16 and 1/4 second, then this is:

(N² / t) = (16 × 16 ÷ 1/4) = (16 × 16 × 4) = 1024.

Log₂(1024) is EV 10. Meaning, 2¹⁰ = 1024.

EV is the exponent of 2 that computes the value, again meaning 2¹⁰ = 1024 value. I don’t know the units of this exposure 1024, if it has any, but it represents the settings combination of (N² / t) and corresponds to proper exposure of the ISO and the light level, in terms of camera settings.

Important: Note that f/stop values must first be squared in exposure calculations, see squaring just below.

But several other combinations of (N² / t) numbers could also compute 1024 (Equivalent Exposures). The purpose is that EV is an exposure scale in exact steps of 2x photo stops, which are seen in a EV Chart , where a row lists the Equivalent Exposures for that exposure (of ISO and light level). Meaning, for any ISO used, if you meter the light as EV 10 at that ISO, then the correct exposures are in the EV chart row for EV 10.

The basis of EV = log₂(N² / t) is that 2^EV = N²/t. EV is the exponent to compute the steps of powers of 2, and Log₂(value) is value expressed as a power of 2. EV is that exponent of 2. The 2 means that one EV is 2x more light which requires 1/2 the exposure. Clearly greater f/stop Number, or decreased time is less exposure and greater EV number, and 2¹ measures steps 2x or 1/2x. Full stops of shutter speed or aperture affect exposure in powers of two. The precise 2x steps are the entire purpose of EV.

The starting point of the EV numbering is that f/1 at 1 second is EV 0. This is only because 1² / 1 = 1, so that log₂(1) is 0 EV, and also 2⁰ = 1 EV. It is math, and not some human notion of convenience (as are the Nominal value markings). However, the camera goal is to instead use the precise values so that every EV step is exactly 2x exposure.

Necessary details to know first: Both shutter speed and f/stop are marked with nominal approximate values established by about 100 years of convention (Nominal meaning those exact numbers do not necessarily actually exist due to them just being approximations). Back then, the binary numbering sequence of 1, 2, 4, 8, 16, 32, 64, etc. was not as mainstream as today. Modern cameras are computers that instead actually strive to use the specific and precise binary numbering goals (exact powers of 2) to ensure precise 2x exposure steps called EV. For example, the value f/5.6 is just a nominal approximate number used for simpler camera marking. Its precise value in math is f/5.65685 (because it is the 5th power of √2). And 1/250 second is actually 1/256 second. No claim is made here about actual camera mechanical accuracy (which is quite good today), but these are instead the exact PRECISE GOALS of design, the sequential binary powers of two. They are the intended numbers computing should use for precision. And accuracy is expected to be quite good today, compared to the early past.

If any doubts, time your camera’s 30 second shutter speed, and you will see it takes exactly 32 seconds (or 15 seconds takes 16). Those are the exact numbers of the powers of 2.

√2 is a key factor for the circular f/stop aperture because the area of the circular aperture is Pi × r². Doubled area is 2 × Pi × r² = Pi × (√2 × r)² Multiplying the aperture radius by √2 doubles its area and exposure. The value of √2 is 1.41421.

F/stop Number = focal length / (effective) aperture diameter. Full stop f/stop Numbers are integer steps of powers of √2, which numbers are proportional to the area of the circular aperture. The purpose of using this f/stop numbering definition is that then any camera and lens using f/8 is the same exposure as all other cameras at f/8 (if also at same ISO and shutter speed). The idea of EV is that ± 1 EV is exactly 1/2x or 2x the exposure, and to be precise, calculations need to use the precise setting numbers to compute EV precisely.

A very important quirk is that most further calculation of exposure (like EV) must first square f/stop numbers (or the ratio of f/stop numbers are reversed and squared) to represent exposure (except NOT for Guide Number, which already takes this into account). Because the equations want values of exposure which is powers of 2, but f/stop Numbering steps are powers of √2. But squaring the f/stop number becomes the necessary steps of powers of 2. Shutter speeds and ISO are already powers of 2 and are Not squared.

To compute precise values, the value called Stop Number is the sequential numbering of the stops, simply numbered as 0, 1, 2, 3, 4, etc. This is the exponent of 2 (or of √2 for f/stops) to compute the exposure value.
The scale starts with Stop Number 0 at value 1, because any number to power of 0 is value 1.
Negative exponents compute fractional values less than 1 (Example: 2³ = 8, 2^-3 = 1/8).
Logarithms of fractions (less than 1) are negative (the logarithm is the exponent of the base).

Stop Number and f/stop Number are Not the same things.
The precise goal value for f/stop Number is (√2)^{Stop Number} (so that steps of the exposure values will still be the powers of 2).
Example: (√2)⁷ = f/11.314, which we simply call f/11.
F/stop value can also be computed as square root of (2^{Stop Number}) .

f/stop (aperture)
Nominals	f/0.5	f/0.7	f/1	f/1.4	f/2	f/2.8	f/4	f/5.6	f/8	f/11	f/16	f/22	f/32
Precise Goal	0.5	0.707	1	1.414	2	2.828	4	5.657	8	11.314	16	22.6	32
Stop Number	-2	-1	0	1	2	3	4	5	6	7	8	9	10

The precise goal value for shutter speed is 2^{Stop Number} seconds (which steps are the powers of 2).
Example: 2^-7 = 1/128 second

Shutter Speed (duration, seconds)
Nominals	30	15	8	4	2	1	1/2	1/4	1/8	1/15	1/30	1/60	1/125	1/250	1/500	1/1000
Precise Goal	32	16	8	4	2	1	1/2	1/4	1/8	1/16	1/32	1/64	1/128	1/256	1/512	1/1024
Stop Number	5	4	3	2	1	0	-1	-2	-3	-4	-5	-6	-7	-8	-9	-10

The approximated Nominal values are close enough for human thinking, but are not precise enough for computing. Nominal values are just various arbitrary approximations established by years of convention, and so their values cannot be computed. Therefore the precise goal values cannot be computed from the Nominals. Nominals are the name we call them, but the Precise values are the actual exact precise design goal, computed to be exact powers of 2 from the Stop Number sequential order, and are the goal targets that the cameras strive to achieve. The f/stops are mechanical mechanisms, but today, camera shutter speeds are controlled by a digital computer chip and a quartz crystal chip, like a digital watch. Which also keeps time and date for your digital camera.

The “every other” or full f/stops of f/1, 2, 4, 8, 16, etc are actually the precise goals (actual powers of 2), and the shutter speed full stops from 1/8 second to 8 seconds are also the precise values. The rest of the nominals are approximations, more or less rounded some way for humans, but the camera still does it right.

Each of the full f/stop numbers (1, 1.414, 2, 2.828, 4, etc) is a √2 multiple, which when squared (N²) is a sequential power of two, which is why the EV formula uses N². Shutter speeds are similar Stop Number calculations, but already using powers of 2 instead of √2. And negative stop number exponents compute fractional values (1/2 second or f/0.5 for example).

See charts of these setting values at precise goal values.

For any fractional stop (third, half, or tenth stops), just add the fraction to the base stop number. One example third stop is nominal f/5.6 + 2/3 EV, which for Stop Number 5 is (√2)^(5 + 2/3) = 1.41421^5.66667 = f/7.1272. A couple of extra significant digits are good for precision for exponents (Wikipedia Significant digits). So for exponents, let it compute the 2/3 or use 0.66667 instead of 0.67. Especially for exponent math — note that 1.4⁵ = 5.38, but 1.4142⁵ = 5.656, so for base and exponent, I suggest using another significant digit or two more than you want to see as desired precision (Integers like 2 or 1/2 are OK as is, already precise.)

EV Formulas

The log₂ in the EV formula says that the EV increments in powers of 2 which is steps of 2x exposure. It can be about relative changes in exposure, or any absolute light level relates to a metered light level and ISO (in days of film, that ISO was a constant until we loaded different film). EV is simply a method of counting stops of change of exposure. But once associated with a proper metered exposure with ISO, EV can also become a meaningful absolute number of exposure.

The EV Chart has already have done the math, and the camera automation does it for us. However, some want to know details.

2^EV = (fstop² / shutter time)

EV = log₂ (fstop² / shutter time)

Points about photographic EV ratios:

The f/stop NUMBERS are √2 steps. The EXPOSURES are 2x steps, but the f/NUMBERS are √2 steps, so again, any calculations must first square f/stop numbers (or the ratio of f/stop numbers are reversed and squared) to represent exposure. (But DON’T square shutter speeds or ISO).
To compare f/11 to f/2.8 (which we know f/11 is -4 EV from f/2.8)
(f/stop ratios are reversed division due to f/stop numbers running backwards)
n = 2.828/11.314 = 0.25 (with precise values of f/stops)
EV = log₁₀(n²) / log₁₀(2) = -4 EV difference (log of fractions are negative)
Or n = 11.314/2.828 = 4 f/2.8 compared to f/11
EV = log₁₀(n²) / log₁₀(2) = +4 EV difference
But shutter speed and ISO are powers of 2, and run forward, so :
To compare 1/30 to 1/125 (which we know 1/30 is +4 EV from 1/125)
n = (1/32) / (1/128) = 128 / 32 = 4 (with precise values of shutter speeds)
EV = log₁₀(n) / Log₁₀(2) = 4 EV difference.
Shutter speed and ISO numbers and ratios are Not reversed or squared, only f/stops.
The log₂ is needed to compute EV because the basic concept is 2^EV = N²/t EV is the exponent of 2 that computes the value. 1 EV is 1 photographic stop = 2x light value = 0.5x exposure necessary.
Many camera setting numbers are just nominal approximations, close enough for human interest, but their numbers only compute approximations (but the Precise values work precisely in every case). See the Nominal/Precise charts of Camera Numbers.
Nominal f/5.6 is actually f/5.657 and nominal f/11 is actually f/11.314. Repeating with the difference of these approximate ratios is:
Nominals: n = 11 / 5.6 = 1.964 (not 2.0)
Squared is 3.858 (not 4.0)
EV = log₁₀(n²) / log₁₀(2) = 1.94 EV from nominals.
But these nominal values don’t actually exist — the camera uses the precise numbers.
Precise: n = 11.314 / 5.657 = 2 (precisely 2x for 2 stops, and 2² is 4)
EV = log₁₀(n²) / log₁₀(2) = 2 EV precise value — This is log₂ (n²)

EV is the exposure effect of the light that the camera settings see. Then ISO is a sensitivity which matches those camera settings to the scene light level, which makes ISO also be pretty important. The EV formula does not use ISO directly, however the camera settings that we choose and put into the formula definitely do depend on ISO. EV is the light level matched by the ISO in the camera settings we choose.

Mathematically, the EV values are exact only if computed with the actual precise theoretical settings that the camera uses, as opposed to the camera’s nominal marked values. The EV calculator converts nominals to the precise values. For example of the nominal marked numbers vs the precise theoretical numbers, f/11 at 1/60 computes EV 12.826. However using the actual real values of f/11.314 at 1/64 computes exactly EV 13.00 (the correct value). Techie details maybe, but that's how it works.

EV is the power of 2 that equals N²/t. N is f/stop Number, t is time duration. Then ISO shifts that value appropriately.

Relative EV: A dim exposure of half the sufficient amount needs +1 EV more of settings or ISO to correct it.

Absolute EV: Twice the light intensity or twice the ISO meters +1 EV higher, and requires half the exposure.

EV might seem reversed relative to exposure settings, because metered higher EV is brighter light or greater ISO, which requires settings for less exposure.

ISO Conversion of EV

EV_S = EV₁₀₀ + log₂ (S / 100) where S (speed) is the new ISO value (converted FROM EV₁₀₀, this from Wikipedia).

A light meter will do it this way to convert the reading to the ISO value you use. ISO 100 is not required by EV, the only thing special about ISO 100 is that it is a familiar conventional value. It could be any numbers, for example conversion TO EV_S at ISO S FROM EV₈₀₀ at ISO 800:

EV₈₀₀ converted to ea New ISO S = EV₈₀₀ + log₂ (S / 800) = EV_S

Tricky part: Converting from higher ISO to lower, or from lower ISO to higher, cannot both add the difference, one must subtract. But if the ratio of the Numerator/Denominator fraction is less than 1, the log becomes negative, so the formula's final ± sign is determined by the log of ISO2/ISO1. One of ISO2/ISO1 or ISO1/ISO2 will be a fraction less than 1 (Or else if equal, log(1) = 0). So this is saying that the NEW ISO must be the numerator (on top).

If S is to be ISO 1600, then 1600/800 is 2, and log₂(2) is +1, so it adds 1 EV.
If S is to be ISO 400, then 400/800 is 0.5, and log₂(0.5) is -1, so it adds -1 EV.
There is a logarithm section just below.

Light Value

EV₁₀₀ is the Conventional Definition of Light Value. Light Value is absolute EV converted to ISO 100, used as a standard comparison of exposures of actual scene light values. The convention is that EV at ISO 100 can be numerically compared for exposure difference or scene brightness.

The Wikipedia link above doesn’t concede the convention of the term Light Value being ISO 100. However, I believe it obviously is the common convention. For example, ISO 100 is what they use in their page’s Lighting Condition chart, and ISO 100 is also used in ExifTool Light Value. Light Value of ISO 100 may not be a rule or law, but it is the common convention. However, Wikipedia does conclude by showing this final version of EV₁₀₀ from EV_S, which common convention and I name to be used for Light Value:

EV₁₀₀ = EV_S - log₂(S/100)

It is just a regular ISO conversion, in this case TO ISO 100. If using Minus, the NEW ISO has to now be the denominator (on bottom), for subtracting. Or alternatively, instead reverse the numerator/denominator and the fraction will reverse the sign, but DO NOT do both). The EV calculator and the one on the Exposure Comparison page compute Light Value by converting **TO** 100, which does use minus, like this:

Light Value (as converted to ISO 100) = EV₁₀₀ = EV_ISO - log₂ (ISO / 100)

The term Light Value arbitrarily uses EV at ISO 100 to specify the scene light level, as a comparison standard of scene light value, but ISO 100 itself might not always allow usable (selectable or desirable) camera settings for that exposure. But it is still an Equivalent Exposure that can compare exposure EV levels.

The purpose of Light Value is that comparing two exposures as both using the same ISO (and ISO 100 is very familiar and conventional) allows a standard comparison of the two light values. Any value of EV requires asking “at what ISO?”, and ISO 100 is very familiar. This standard convention of converting to ISO 100 is called Light Value (LV), using this additional factor to change to EV for ISO 100.

So Light Value as commonly seen is the convention that means conversion to EV at ISO 100. ISO 100 is not a magic number, and is not necessarily the ISO or settings that the camera actually used. The number 100 is just another number, and is an arbitrary choice, and other than convention is NOT special here. It might sometimes seem special only because the formula arbitrarily chose to use 100. We do commonly use ISO 100, but as Light Value, it is just an arbitrary ISO reference for comparison, and is not trying to imply the settings at ISO 100 are somehow more meaningful than the ISO the camera actually used to get the picture. We can only directly compare absolute EV values if at the same ISO. Which might as well be ISO 100 because we are familiar with it.

Note that Light Value might make it seem like the EV chart must represent ISO 100 because they seem to match, but only because we arbitrarily chose ISO 100 as a reference (and the EV chart matches any ISO actually used). But to make meaning of Light Value, the convention is that “Light Value” does represent ISO 100. Light Value is only used to compare two exposures. The exposure calculator at this site uses Light Value of EV at ISO 100 to compare exposures at the same ISO. Any ISO number would do the same comparison job, so ISO 100 is not a magic number, only a very familiar comfortable number.

Light Value (LV) is the same number as EV at ISO 100, which can make the EV chart seem to be for ISO 100 (because the EV chart agrees with LV at ISO 100). But we are fooling ourself, this is Only because we arbitrarily specifically referenced LV to ISO 100. LV formula could have used any number (but 100 does seem convenient). The EV chart in fact matches whatever ISO we may be using at the time. EV is computed from the camera settings, which are chosen for the ISO we actually are using.

EV and Logarithms (if you're planning to do the math)

EV is important because a one stop change, which is 2x intensity, is called 1 EV (Exposure Value, more here). Or also a 1/2× intensity relationship, which is -1 EV. EV is the powers of 2 relationship of a light intensity ratio.

EV involves logarithms. We have seen logarithms in high school math. Logarithms were an extremely important tool in history before computers, because manual multiplication and division of big difficult numbers could be instead done by adding or subtracting logarithms (which is also the basis of slide rules, which are fast, but slide rules typically only offer maybe 3 significant digits). Logarithm tables are just lists of numbers with their exponents of the base which correspond to the value. The same log table is also used for look up of antilogs (the values corresponding to the logs). Calculators do this job today (not the basic 4 function models, but the "scientific" models do, and also the Windows CALC app and Javascript, etc).

What is a logarithm?

The definition of logarithm is that it is the exponent representing the power to which a fixed number (the base, like 10 or 2 or e) must be raised to produce a given number.

log₁₀(8) = 0.90309 the logarithm of 8, base 10
10^0.90309 = 8 base^exponent is the value again, called antilog of the logarithm
In base 10, 10¹ = 10, 10² = 100, 10³ = 1000 (each step 10x value)

log₂(8) = 3 the logarithm ( log₂(X) = log₁₀(X) / log₁₀(2) )
2³ = 8 base^exponent is the value again, called antilog of the logarithm
In base 2, 2¹ = 2, 2² = 4, 2³ = 8 (each step 2x value, important in cameras)

Logarithm and exponent are just reverse operations, opposites. We know that addition/subtraction and multiply/divide are also reverse or opposite or inverse operations. Meaning, one can be used to "undo" the other, so to speak.
Log and exponent are also the similar inverse relationship.
So 2³ = 8, and log₂(8) = 3 are inverse operations.

So log₂ of X just gives the exponent of 2 (which here we call EV) that will give X (as powers of 2). So regarding photographic exposure calculations, our exposure in powers of 2 = X are just log₂(X) values. And the log₂ of ratio of two values is also the EV difference. Example ratio 16/2 = 8. log₂(8) = 3, meaning 2³ = 8;

Let's say this last 8 represents the power of a flash, or the intensity of any light. The 8 means relatively, the light is 8x brighter than some other light level that we call 1.
Relative to the other light, 2³ = 8, means in 2x steps, our light is doubled 3 times, or 8 times more light or power, because 2³ = 8. So log₂(8) = 3, which is that exponent of the base 2. And photo EV is about these 2x steps of intensity. 1 EV is 2x value, 2 EV is 4x, 3 EV is 8x. EV is computed as EV = log₂(intensity ratio).

From the 1600s until the later parts of the 1900s (before computers, the time basically from John Napier and Newton to even Einstein), complex math was done with logarithms by more simply adding and subtracting log values, instead of tedious long multiplication/division of big numbers. The logarithm values were looked up in log tables (a book of log values), typically from 4 to 8 significant digits, which showed the precomputed logarithm values. Calculators do it today. John Napier (1550-1617, Scotland) was not the actual inventor of logarithms, but he named and popularized the use of them as computing tools, and he is the name we associate with logarithms.

Just for fun, a simple problem to make the point about the forgotten usefulness of logarithms before calculators:

The value of Pi is 3.14159 (and this log₁₀ is 0.4971495)
If the radius of a circle is 122.125 (and that log₁₀ is 2.0868046)
then the area of the circle is Pi r² = 3.14159 × 122.125² = 46855.3 (log₁₀ is 4.6707587)

Using the logs, it is 0.4971495 + (2 × 2.0868046) = 4.6707587
(multiply logs by the exponent, which if complex, that multiply could be yet another log addition)
Inverse Log(4.6707587) = 46855.3 area of the circle. The data had 6 significant digits.

If that might seem a complication, then to appreciate it, try doing Pi r² = 3.14159 × 122.125² by multiplying with only a pencil. And then contemplate some bigger problem. 😊 The log just came from a table lookup, and then adding or subtracting many digits is much easier than multiplying or dividing them. Then early on (17th century) the slide rule put the logarithms on a sliding stick (adding or subtracting length from another stick), which was easy to use, but was only precise to at most about 3 significant digits (but often adequate, except for banking). But yes, today we sure can appreciate our calculators.

How to compute log₂ (logs of base 2)

EV = log₂ (fstop² / shutter time) (Using precise goal values of settings, NOT nominals)

To compute log₂, then log₂(X) is log₁₀(X) / log₁₀(2)

Or log₂ is log_e(X) / log_e(2) (can use any base, doesn't matter so long as consistent.)

Caution about Programming Languages:

log₂(X) is log₁₀(X) / log₁₀(2)
and log₁₀(2) = 0.30103, so we might see shortcuts just using 0.3
log₂(X) is log₁₀(X) / 0.3, or
log₂(X) is 3.3 × log₁₀(X) (because 3.3 = 1/0.3)

which are fine approximations, but... you should know that programming of the log() function in Excel and Javascript and C and Python and other programming languages do use Natural log base e instead of Common log base 10. Which normally doesn't matter, the results are the same if consistently using same base. However these numerical constants (of 0.3 and 3.3) will give the wrong answer unless using log₁₀ (only due to the constants, because log₁₀(2) = 0.30103 but log_e(2) = 0.693147). So using 0.3 for log_e(2) is simply wrong. So, in the programming languages (which use log_e), just use the log(X)/log(2) instead of 0.3.

Also important, note that these programming languages will also expect trig angles to be radians instead of degrees. Radians = Degrees x Pi/180.

Handheld calculators normally offer log₁₀ processing at button log, and log_e at button ln or lnx (the n meaning Natural and log_e). To compute log₂(X) in base B is log_B(X) / log_B(2).

On a Texas Instruments calculator, this key stroke sequence for log₂(X) is: (the value of X) log ÷ 2 log =

Examples of log bases:
log₁₀(2) = 0.30103
log₂(2) = 1
log_e(2) = ln(2) = 0.69315 (JavaScript returns this constant with Math.LN2 (so use that in Javascript instead of 0.30102)
Using any log base computes correctly if you are consistent, but NOT if you specify a wrong constant assumed for the log base used.

Handheld scientific calculators (and the Windows Calc program) have both modes, and likely have one key called lnx or ln which is Natural log_e, and another key called log which is Common log₁₀, which simply does that conversion. Using either log system works if consistent, but my point here is that in programming languages, log_e(2) is NOT 0.30103.

Again, here's the necessary trick about computing EV ratios of values. Stops of shutter speed or ISO are 2x steps of intensity, but the literal f/stop Numbers are √2 steps. We know their meaning, but the actual f/stop Numbers must be squared to be proportional to 2x steps of Exposure. Squaring converts the √2 factor to a power of 2, necessary for EV and any other exposure calculation. Most exposure calculations (Except Not for Guide Number) simply must first square f/stop Numbers or f/stop Ratios (to become powers of 2 instead of powers or √2 Squaring is NOT done for shutter speed or ISO intensity ratios. And ratios of f/stop Numbers are reversed for proper sign, and squared. Squaring f/stop NUMBERS was said several times now, I hope it is clear.

Examples of ratios:

The difference in ISO 400 from ISO 100 is log₂(400 / 100) = 2 EV (± result is the first relative to the second).
The difference in 1/4 second from 1/16 second is log₂(0.25 / 0.0625) = 2 EV.

The difference in f/8 from f/4 is log₂( (4 / 8)²) = -2 EV. (f/stop ratios must be squared first)
The difference in f/11 from f/5.6 is log₂( (5.657 / 11.314)²) = -2 EV.
(precise values compute precise results, but Nominal log₂((8 / 11)²) = -0.919 (11 is an approximation)

Or the ratio of 8 seconds vs. 1/8 second: Logs of fractions less than one are negative (because negative exponents create fractions), so log₂(8) = 3, and log₂(1/8) = -3. In APEX terms (subtraction of exponents is same as division of the values), so this ratio difference is EV = 3 - (-3) = (3 + 3) = 6 EV. Or EV is log₂ of the ratio of two intensity values, log₂(8 seconds ÷ 1/8 second) = 6 EV difference (and 2⁶ = 64 exposure difference factor).

Smaller numbers are less time exposure or less ISO exposure, but are greater f/stop exposure (running opposite directions). To handle that, I simply reverse the division order for f/stops, as shown. Fractional ratios are negative logs. You will compute precise results (for example, for compensations or EV values or Guide Number values) if you use the precise values instead of the approximate nominals.

EV is the power of 2 that gives this intensity ratio, 2x multiples, of 1, 2, 4, 8, 16, 32 ... values of intensity. Log₂ computes that exponent of that 2.

EV sign of ratios of values is negative if ratio is a fraction less than one.
And the EV result is an exact precise value if you use the exact precise setting values.

Logarithms are used for many common purposes (described as powers of 2 or 10 or e, etc):

Photographic exposure increments: 2^EV = N²/t so EV = log₂(N²/t)
Ph scale of acidity in chemistry: Ph = -log₁₀(hydrogen ion concentration). Distilled water is Ph 7.
Interest rate and biology (growth calculations, log_e).
Wind speed and earthquakes and audio volume dB (intensity, log₁₀)
88 key keyboard music scale note frequency is a 12 note octave based on 12th root of 2.
With standard of A₄ being 440 Hz, Nth piano key frequency = 2^(N-49)/12 x 440 Hz.
And A₃ will be 220 Hz and A₅ will be 880 Hz.
key 1 A₀: 27.5 Hz key 40 Middle C₄: 261.63 Hz Key 49 A₄: 440 Hz Key 88 C₈: 4186 Hz
Before computer chips, manual multiplication and division of large numbers could be done easier by adding and subtracting logarithms on paper. Or exponents of a value were actual multiplied or divided with the logarithm value. Or slide rules also worked by adding and subtracting distances on logarithmic scales (using lengths on the sticks marked with logarithmic scales). Possibly this is very foreign today, but logarithms were the computing methods for about 300 years, until transistor calculators about 50 years ago. But until then, slide rules were dear to the heart of very many people, including all engineers and all engineering students.
There were several scales on the slide rule, like for trig functions and square and cubes or roots, and more, but the main ones for multiplication and division were the C scale (top below) and the D scale (bottom below) Numbers had to be reduced to be between 1 and 10, and decimal points readjusted by hand afterwards, with additional rules.
Slide rule example: 5.5 ÷ 2 = 2.75 (same method if 5,500,000 / 0.0275)
which is log(5.5) - log(2) = log(2.75), or if instead done on paper (with log tables),
then is 0.74036 - 0.30103 = 0.43933, which antilog(0.43933) can be looked up in log tables, or antilog is 10^0.43933 = 2.75
On the slide rule shown below, Log(5.5) - log(2) is 2 on the top C scale aligned with 5.5 on lower D scale, and the subtraction result is at 2.75 on D (at the 1 on the end of the C scale). For readability, you could move a line on a clear window cursor over the 5.5, which then also was over the 2 mark on the C scale.
Or a multiplication addition result is shown on the current slide rule position as 2.75 x 2 = 5.5.
Some numbers like 2 x 9 would not line up as shown, so the top C scale had to extend out of the left end, and the right hand 1 index (10) was used. That indicated 1.8.
Precision is harder to read precisely at the right end (more than 2 significant digits was arguable), so results were reasonable approximations, not exact.

Calculating Precise Camera Setting Values with Stop Number

An Introduction to the “Precise goal” calculations is above.

The number of significant digits (of every factor) is important in computing. Modern cameras are pretty accurate, but my using many digits in the chart results does NOT imply mechanical hardware accuracy is always quite that precise. But the target goal is very precise. The charts above may "show" Stop Number here with fewer digits (shown with minimum of four here), but it is an exponent (has large effect), so Stop Number is always used as its full actual precise fractional values (used as like 2 + 1/3, which computes as Stop Number 2.333333) for adequate precision of the exponent math. More significant digits of stop number to needed to compute and round accurate large full numbers for 1/16384 second, or ISO 1032127 (which require more than 2 significant digits). Saying, if your own calculation wants to see five accurate significant digits, then all the numbers you use to compute it must have at least five accurate significant digits too. (I find one additional digit works well). But a value like integer 3 if like in "3 dogs" is of course a fully precise value called an Exact Number, see Wikipedia Significant digits.

So the way to compute the precise powers of 2 is to simply number the nominal setting values consecutively, as 0, 1, 2, 3, 4, 5, 6 ... called Stop Number (if that consecutive numbering is not clear here, see the larger example above). These will be the exponents of 2 or √2 computing the precise powers of 2 (to ensure each step is exactly 2x exposure).

Shutter speed is computed as base 2 to exponent Stop Number. But the nominals in the text can appear their nominal way.
F/stop is computed as base √2 = 1.4142 to exponent Stop Number.
Stop Number 0 computes the value 1 (as f/1 or 1 second) because any number to power 0 is 1.

Fractional stops are computed with exponent (Stop Number + fraction).
Example: f/5.6 + 2/3 EV = (√2)^{(5 + 0.666667)} = f/7.12719

f/stop (aperture)
Nominals	f/0.5	f/0.7	f/1	f/1.4	f/2	f/2.8	f/4	f/5.6	f/8	f/11	f/16	f/22	f/32
Precise Goal	0.5	0.707	1	1.414	2	2.828	4	5.657	8	11.314	16	22.6	32
Stop Number	-2	-1	0	1	2	3	4	5	6	7	8	9	10

The precise goal value for shutter speed is 2^{Stop Number} seconds (which steps are the powers of 2).
Example: 2^-7 = 1/128 second

Shutter Speed (duration, seconds)
Nominals	30	15	8	4	2	1	1/2	1/4	1/8	1/15	1/30	1/60	1/125	1/250	1/500	1/1000
Precise Goal	32	16	8	4	2	1	1/2	1/4	1/8	1/16	1/32	1/64	1/128	1/256	1/512	1/1024
Stop Number	5	4	3	2	1	0	-1	-2	-3	-4	-5	-6	-7	-8	-9	-10

EV calculations require logarithm math detailed above.

The precise goal value is this small formula to calculate the precise value:

Value = Base^{Stop Number}

A Few Examples
Stop Num	f/stop		Shutter Speed
Stop Num	Precise	Nominal	Shutter Speed
-2	f/0.5	f/0.5	1/4 sec
-1	f/0.707	f/0.7	1/2 sec
0	f/1	f/1	1 second
1	f/1.414	f/1.4	2 secs
1 1/3	f/1.587	f/1.6	1.5 secs
1 2/3	f/1.782	f/1.8	3.2 secs
2	f/2	f/2	4 secs
3	f/2.828	f/2.8	8 secs
4	f/4	f/4	16 secs
5	f/5.657	f/5.6	32 secs
5 1/3	f/6.35	f/6.3	40 secs
5 2/3	f/7.127	f/7.1	51 secs
6	f/8	f/8	64 secs
7	f/11.314	f/11	128 secs
2^StopNum is shutter speed 2³ is 8 seconds 2^-3 is 1/8 or 0.125 seconds
(√2)^StopNum is f/stop number 1.414³ is f/2.828 (nominal f/2.8)

For shutter speed and ISO, the base is 2, to match the 2x stops of EV values. But due to the √2 steps of f/stop being 2x stops, f/stop alone uses base of √2.

The concept is there are "Stop Numbers", simply numbered 0, 1, 2, 3, 4, 5... starting at Stop Number 0 which calculates a value of 1. We did not just arbitrarily number them, but instead these are the math exponents, to create powers of 2. Any base number to these exponents creates intervals of the exact powers of that base, and base 2 gives powers of 2 used for Shutter Speed and ISO. 2^Stop Number is an exact power multiple of Base 2.

Stop Number 0 is the starting point, because any non-zero number to the power of 0 is value 1 (0⁰ is not defined). Stop Number 0 starts the all-important value sequence 1, 2, 4, 8, 16, 32, etc. (a base of 2 computes powers of two). Photography is concerned with 2x exposure steps, so our base is 2 (except f/stop numbers use a base of √2, because our lenses do). So starting points are that 2⁰ is 1 second, and (√2)⁰ is f/1.

Fractional partial stops (like third or half or tenth stops) simply add the fraction to the stop number to compute the value of Value = Base^{(Stop Number + fraction)}.

Note that "Stop Number" is the simple numbering of "stops", 0,1,2,3,4, etc. When used as the exponent of the base, it computes the "f/stop number" or the shutter speed values. Negative exponents create fractions, for example 2^-2 = 1/4 second, or (√2)^-2 = f/0.5. The numbers are necessarily what they are because of this math of the powers of two.

A few examples of concept are shown at right, but Stop Number is also shown in the long charts just above. The Stop Numbers are not shown in the camera, but they are used to compute the camera setting goal targets that are used. The Stop Number as an exponent for thirds may need 4 or 5 significant digits in your calculator (I use 1/3 in programs, don't use just 0.33) to compute a value this close, and then you can round it as desired.

Implementation of Stop Number

I don’t intend to try to teach math or coding, but do hope to explain my material about camera basics. If you’ve read this far, you’re surely interested (and me too), so warmest welcome, you’re my target audience. This part is “how to do it”, and it might be of interest.

Select Box HTML values

Shutter Speed HTML Sample
<option value=-1/3>1/2.5
<option value=-1/0>1/2
<option value=-0/7>1/1.6
<option value=-0/5>1/1.5 *½
<option value=-0/3>1/1.3
<option value=0/0>1 second
<option value=0/3>1.3
<option value=0/5>1.5 *½
<option value=0/7>1.6
<option value=1/0>2
<option value=1/3>2.5

f/stop HTML Sample
<option value="4/7">f/5.0
<option value="5/0">f/5.6
<option value="5/3">f/6.3
<option value="5/5">f/6.7 *½
<option value="5/7">f/7.1
<option value="6/0">f/8
<option value="6/3">f/9
<option value="6/5">f/9.5 *½
<option value="6/7">f/10
<option value="7/0">f/11
<option value="7/3">f/13

Nominal values cannot be computed, and the precise goal values cannot be computed from Nominals, because Nominals are just various arbitrary approximations determined by convention. However, the precise values are the camera designers goal, and the precise values are well known, because their serial order of full stops is still there, and each full stop is a 2x exposure change. F/stop NUMBERS are √2 steps, but are still 2x EXPOSURES. The precise powers of 2 or √2 are created by the Stop Number being the exponent of the base, simply determined by the 1,2,3,4 sequential order of the stops.

So for HTML and JavaScript, my notions were to code my settings HTML Select boxes as shown in this list. The list box visible Text shows the nominal convention, and the Value like 6/3 is a code that means Stop Number 6 (some of which are negative, in which case an increasing fraction should retain the minus sign too). This example /3 is the fraction (where 0 means Full, 3 means 1/3, 5 means 0.5 and 7 means 2/3). Then the code 6/3 can compute the precise value like 2^{(-6 -1/3)} which is 2^(-6.333333). Or a f/stop code might be 5/7, meaning √2^{(5 + 2/3)}. This method just calls a function (below) to compute that precise value. The Select lists could instead be hardcoded with the precise values, but this method makes calculations of new values be easy (EV simply adds to the final processed Stop Number).

ISO adds an offset 0.643856189 to Stop Number (in the code, but it’s not in the Select box), which is log₂(100/64), for the purpose that instead of the old code 0/0 (= ISO 1), ISO now begins at 0.643856189/0 which is ISO 1.5625 (APEX method, all below). APEX actually starts at ISO 3.25 at log₂(32/100) = 1.643856189/0, but I sometimes preferred to also show it as 1.5625 (and there is no other change). This procedure makes ISO 100, 200, 400, etc be exact Full stops, meaning with the third stops computed relative to those values. Frankly, if we don’t mind if 1/1000 second is actually 1/1024 second, and f/11 is actually f/11.314, I don’t know why we care that nominal ISO 100 was 101.6. It is 0.023 EV difference, but it’s convention now.

This might seem a bit complicated, but it is the definition of the stop numbers, and it’s actually the easiest and most versatile way to be precise. Here’s some JavaScript help which I’ll share to show the concept. There are a few ways to make errors, so an understanding would be good before starting. My way is this:

Stop Number conversion which returns precise value from the Select code.

function precise(which, code) {
/* StopNumber conversion, returns precise setting value
which is "s", "f", or "i" for shutter speed/ fstop or ISO.
code is the "-4/7" format from SELECT, to become stop number -4.666667
If which is UpperCase, that returns stopnum instead of value (for further calculations).
ISO select boxes do not have ISO offset code in them, but this adds it.
*/

var a2, stopnum, fraction, base, return_stopnum;
if (typeof code === "number") //accept formats text 2/3 or number 2.333333
stopnum = code;
else {
a2 = code.split("/");
stopnum = Number(a2[0]); //so can add them below
fraction = (a2.length > 1)? Number(a2[1]) : 0;

if (fraction == "3") fraction = n13; // = 1/3; //use computed fraction for adequate significant digits
else if (fraction == "7") fraction = n23; // = 2/3;
else if (fraction == "5") fraction = 0.5;

stopnum += ((code[0] == "-") ? -fraction : fraction) //look for negative stop number, add fraction to int portion
}

return_stopnum = which < "a"; //uppercase? return stopnumber only?
base = 2;
if (which == "f" || which == "F")
base = Math.SQRT2;
else if (which == "i" || which == "I")
stopnum += ISO_offset; // = Log(100/64)/log(2) = 0.643856189 added to make ISO 100 be a full stop

return (return_stopnum)? stopnum : Math.pow(base, stopnum);
// return stop value, vs return stop number as 2.333333

}

Shutter Speed Values

Shutter Speed Exponent is Stop Num
2^N	Duration
2⁵	32 sec.
2⁴	16 sec.
2³	8 sec.
2²	4 sec.
2¹	2 sec.
2⁰	1 sec.
2^-1	1/2 sec.
2^-2	1/4
2^-3	1/8
2^-4	1/16
2^-5	1/32
2^-6	1/64
2^-7	1/128
2^-8	1/256
2^-9	1/512
2^-10	1/1024
2^-11	1/2048
2^-12	1/4096

Shutter speed values are base 2 to power of stop number — numbering is the powers of 2 so their exposures are exactly 2.0x apart (1 EV). Third stops are cube root of 2 apart, and three thirds add to 1.0 EV. Base 2 to power of Stop Number 3 is value 8 (seconds). Base 2 to power of Stop Number -3 is value 1/8 (seconds).

All full stops are precisely 1.0 EV apart (2.0x intensity). This is a sacred rule, and powers of 2 ensure this precise goal. Doubling shutter time duration or the ISO value is 1.0 EV (powers of 2). However doubling any f/stop number is -2 EV (which is sometimes a handy thing to know), due to being powers of √2 (next below).

This method computes the theoretical "precise" goals actually used, which are often slightly different than the nominal numbers we see marked. The nominal marked numbers are just a convenient rounded guide for humans, but the digital camera design always uses the precise values. The mechanical result could still vary slightly, but modern digital camera timing is much more accurate than in the past (before digital).

Stop Number creates full stops (of base 2) in a binary 1, 2, 4, 8, 16, 32 ... sequence. The need for 2x exposure steps seems obvious. Stop Number itself is 0, 1, 2, 3, 4, 5 ... for full stops (stop number is the exponent calculating either powers of 2 for shutter speed, or powers of √2 for f/stop). Any number (except 0) to exponent 0 is 1.

The value 2^Stop Number computes the 1, 2, 4, 8, 16, 32 ... shutter speed full stops. 2x shutter speed duration is 2x exposure (linear), which is one stop. The base is 2⁰ = 1 second.

Negative Stop Numbers will compute fractional numbers (less than one, like 1/4 second).

For fractional stops (like third or half stops), add the fraction (0.333333, 0.666667, or 0.5) to the stop number.

F/stop Values

f/stop = focal length / aperture diameter, so aperture diameter = focal length / f/stop Number. Aperture is not the actual physical diameter, but is instead the effective diameter as seen though the magnification of the lens front element (see Wikipedia - Entrance Pupil). This definition causes the same f/stop number to be the same exposure on any lens of any focal length (so light meters read the same f/stop regardless of which lens).

The Area of a circle is Pi × r². Doubled area is 2 × Pi × r² = Pi × (√2 × r)² So increasing the aperture radius by √2 doubles its area which doubles its exposure which is 1 EV. Doubled aperture areas increment exposure in steps of EV 1.0 (2x), but the f/stop Number doing that increments in steps of √2. Stop Number is the exponent, and f/stop number increments as √2 × Stop Number. The beginning point is √2 × 0 = f/1. √2 is 1.414, so each full f/stop Number is 1.414 x the previous f/stop number, each of which is a 2x stop of EV exposure change.

Aperture Exponent is Stop Num
(√2)^N	f/Number
1.414⁰	1
1.414¹	1.414
1.414²	2
1.414³	2.828
1.414⁴	4
1.414⁵	5.657
1.414⁶	8
1.414⁷	11.314
1.414⁸	16
1.414⁹	22.627
1.414¹⁰	32

Aperture f/stop values are base √2 to power of stop number (numbering is the powers of √2, so full stop values are 1, 1.414, 2, 2.828, 4, 5.657, 8, etc.) Each full stop value number is exactly √2 apart (1.4142x apart), so that their exposures are exactly 2.0x apart (1 EV). As mentioned before, a handy thing to know is that doubling the actual f/stop number is -2 EV of exposure. Third stops are cube root of √2 apart, and three thirds add to 1.0 EV. Base √2 to power of Stop Number 3 is value 2.828.

For fractional stops (like third or half or tenth stops), add the fraction (0.333333, 0.666667, 0.5, or 0.1) to the stop number.

Negative Stop Numbers will compute fractional numbers (less than one, like f/0.5).

We can compute stop number -3 to be f/0.35, but -2 for f/0.5 is considered a physical limit for a lens to be able to focus well. But f/1.4 or f/2.8 are often a reasonable practical limit (due to diameter, affecting size, weight, quality and cost).

There is a choice for computing f/stop that you might occasionally see. Due to the actual f/stop numbers being multiples of √2, f/stop can optionally be computed as (√2)^{Stop Number}, or as: sqrt(2^{Stop Number}) (same math). In math, when the 2 is moved outside the sqrt radical, it must become √2 out there. Same math, same everything. So FWIW, (√2)³ = 2.828 is the same as sqrt(2³) = 2.828 (seen sometimes).

An Alternative Method

Intervals progress in incremental steps. Note these steps are to be multiplied or divided, Not added or subtracted. A disadvantage is that you must know the precise adjacent step, and the precise step value (as opposed to just easily counting up from zero to the appropriate "stop number").

f/stop: Full stop = 1.41421356× intervals ( √2)
third = 1.122462× intervals (is cube root of √2)
half = 1.189207× intervals (is square root of √2)

Example: f/8 - 1/3 EV = 8 × 1.122462 = f/8.979696
Going the other way would divide instead:
f/8 + 1/3 EV = 8 / 1.122462 = f/7.12719 (these values are in the chart above)
But value 8 is the Stop Number 6 (from 1, 1.4, 2, 2.8, 4, 5.6, 8), so standard way is (√2)^{(6 + 0.3333333)} = f/8.979696

Shutter Speed: Full stop = 2× intervals
third = 1.25992105× intervals (is cube root of 2)
half = 1.41421356× intervals (is square root of 2)

Example: 8 sec + 1/3 EV = 8 × 1.25992105 = 10.079368 sec
or, 8 sec - 1/3 EV = 8 / 1.25992105 = 6.349604 sec
But value 8 is the S.S. stop number 3 (from 1, 2, 4, 8), so the standard way is 2^{(3 + 0.3333333)} = 10.079368

ISO Values

Old ISO Examples
Stop Number	ISO
0	1
1	2
2	4
6	64
6.6667	101.59
7	128
New ISO
0.643856	1.562
1.643856	3.125
6.643856	100.00

ISO is similar to shutter speed (linear powers of 2), except ISO has become special now. ISO film speed before 1975 was called ASA in the USA. ISO still uses the same ASA numbers, which initially began at IS0 1 (from 2⁰). At the time, the European DIN rating also used 2⁰ as a base. Using the standard Stop Number rule that 2⁰ is value ISO 1, and then 2^6.667 = ISO 101.59 which is a third stop less than full stop 125 (like the nominal 1/125 second shutter speed number, both actually 128). Which was OK in film days then, we just bought the film and didn't care if ISO 128 was the full stop.

However modern cameras don't use the slowest ISO numbers any more. And since roughly about 1960 (when light meters started appearing in cameras), have adopted the goal for ISO 100 and its multiples to conveniently be full even stops. So, to make the ISO numbers match current practice, the base for stop numbers is at exactly an full stop multiple under ISO 100. It could use a base 4 or 6 stops under for same result, but ISO choose 5 stops under, then 2⁵ = 32, so that ISO 100/32 = ISO 3.125 is exactly 5 stops under 100. Then log₂(3.125) = 1.643856, and the inverse means that 2^1.643856 = 3.125. So (if this starting point with this stop number offset to reach ISO 3.125), and 5 stops more at 2^{(5 + 1.643856)} is exactly ISO 100 at an even full stop.

If stop number 0 is conventionally to be value 1 (like for f/stops and shutter speeds), then in that system, nominal ISO 100 is stop number 6.6667, which is precise value ISO 101.6, which is a 2/3 stop between full stops 64 and 128.
ISO (called ASA back then) was done that way in say around 1950, when we didn't really know or care about this detail, we just loaded a roll of film. But then in the late 1950s came notions of adding semiconductor light meters into cameras, involving EV computation, and then ISO 100 as being a full stop seemed like it should be convenient for humans, a nice round number.

So to compute this today, we consider ISO 100 to be exactly 5.0 stops above ISO 3.125 (because 100 / 2^5 = 3.125, exactly 5 stops below 100). Then stop number 1.643856, which is log₂(3.125) is ISO 3.125. Then if adding this ISO offset to all stop numbers, then ISO 100 is stop number 6.643956 (stop 5 plus this offset), which is exactly ISO 100.000, which is now a full stop itself (if considered as stop 5).

Or, the math works the same if using base ISO 1.5625 which is 6 stops under ISO 100, so offset in that system is 0.643856. and 100 is stop 6 (same actual 6 + 0.643856 stop number). Same results.

Just compute in the conventional way starting at Stop Number 0 at nominal ISO 3, except add 1.643856 to every stop number, for example 2^{(0 + 1.643856)} = ISO 3.25. The purpose of this plan is that now it now computes precise third and half stops relative to exactly ISO 100 (instead of ISO 101.6 which was previously a third stop). For example, this way, nominal ISO 2500 is a third stop (same as cameras today show it) with precise value 2540. The original way, ISO 2500 was a half stop with precise value 2896.

At the highest end of ISO numbers today, there are no existing nominal values, so the trend seems to be that cameras are forsaking the nominal notation, and began using the actual precise markings (but still no longer based at ISO 1).

That way was simpler, but it's just for explanation why it isn't that way now. It does not change anything except to just shift the numbers slightly. The common values like ISO 100, 200, 400 used to be nominal third stops. They are now precise full stops in today's digital camera.

Nominal values cannot be computed, they're just a list of what convention has called the various stops. Nominals are the same set of numbers in both old and newer charts, but now the third, half and full definitions are shifted. The camera will still find the proper exposure. The nominal numbers are still rounded. Specifically, for the larger numbers, one full stop greater than any nominal third, half or full stop will still be 2x the numerical value.

For empirical example of today's use of this offset, we see a Nikon DSLR if set to ISO 1250 is 1/3 stop less than 1600. The camera then reports ISO 1250 nominal in the menu and Exif, but deep into the extended Exif (Maker Notes section), it also reports ISO 1270 there, that it actually uses. Users don't much care about the exact numerical value, they just want it to be 1/3 stop less than ISO 1600.

The Nikon DSLR also uses 1/6 stops for ISO in Auto ISO mode, so values like ISO 449 or 566 can be seen in that way (called nominal 450 or 560). 566 is 3/6 or 1/2 stop, but 449 is 1/6 stop. This 100/32 = ISO 3.125 starting from 2^1.643856 creates those specific numbers like 1270 or 449 or 566. But starting at ISO 1 does not create the same numbers. Light meters today agree on all the Full and third stop ISO numbers, but I've seen other values from other Auto ISO systems.

APEX Values

This idea of starting ISO at base 3.125 (so ISO 100 would be a full stop) was seen in the APEX (Additive Photographic Exposure) system, which along with EV, was added to ISO specs in the late-1950s when light meters and transistors and batteries started being added into cameras, starting around 1960. That metered exposure calculation required use of the Powers of Two increments.

The APEX idea and math was: EV = Av + Tv = Sv + Bv.

These APEX values are Not camera settings like f/stop and shutter speed seconds, but instead are the Stop Number (the exponents of √2 and 2.
Av is Aperture Value (using for example exponent 3 to denote f/2.8).
Tv is Time Value (shutter speed duration), which these are the exponents (of stop number + fraction), with some changes, for example APEX reverses sign to make Tv be positive (for the additions), but the actual exponent still must be negative for fractional seconds (remember, we didn't have shutter speeds longer than one second back in those days). Some camera brands still label their aperture and shutter preferred modes as Av and Tv, perhaps it indicates it is metered. But these APEX terms are the Stop Numbers (the exponents, not the actual setting values).
Sv is Film Speed exponent, APEX starting at ISO 3.125 at Sv = 0, but still requiring addition of 1.643856 to be the actual exponent (3.125 forces ISO 100 and multiples to be even full stops).
Bv is Brightness exponent (foot candles, starting at 6.25), which the light meter measures to compute EV.

From Wikipedia:
N is f/stop Number, t is shutter speed duration Time.

Assume two nominals, f/11 and 1/125 second. The f/11 value is computed using (√2)⁷ = f/11.314 precise. And the 1/125 second value is 2^-7 = 1/128 precise. So the EV formula is EV = log₂(11.314² / 0.0078125) = EV 14. If instead using the camera nominal values of f/11 and 1/125 second, they would compute EV 13.88, maybe approximately close, but not precise, and Not the actual values used.

In the APEX formula EV = Av + Tv = Sv + Bv, adding the exponents is the same as multiplying the actual values (which is just standard logarithm math, see Wikipedia). The shutter speed -7 is reversed to +7 by APEX. Possible because subtraction of exponents is same as division of the values used by the EV formula (and there were no shutters longer than one second then). But subtracting a negative value is addition again. So then 7 - (-7) = 7 + 7 = EV 14 (for ISO 100), which is the correct EV number. Instead of the EV formula, the APEX addition is much simpler math... 7 + 7 = EV 14 again, by using prepared tables of exponents, necessary because a floating point math processor was not available back in early days of APEX and EV, nor even today in smaller processors.

2⁵ is 32, so ISO 100/32 = ISO 3.125 is exactly five stops under ISO 100. I did start the ISO chart at this Stop Number log₂(100/32) = 1.643856 to be exactly five stops under ISO 100 in order to match APEX charts starting at the ISO 3.125 that you might see. Note the difference though, APEX refers to the ISO 100 "fifth" stop Value as 5, which is the exponent for their additive system of EV, but technically, the math of converting back to specific ISO number still requires the exponent of 2^{(5 + 1.643856)} = 100. The ISO chart here could have started 4 or 6 stops under 100 to still simply match ISO values you may see in cameras today (with identical results as 5 stops), and APEX chose 5 stops to start at ISO 3.25. ISO 100 is just another number, not special in any way, it was just an arbitrary choice, simply a nice convenient number for humans. ISO 128 would have been the natural full stop (2^7), but it computes ISO 100 as a third stop at 101.6 (sort of messy to humans).

The chart values above: To match the cameras, the first ISO chart above places ISO 100 as a precise full stop, and computes correct values for 1/3 and 1/2 stops (correct values relative to ISO 100), and that chart can also show 1/6 stops for ISO too. ISO has not yet defined formal nominal values above ISO 10000, but APEX defines full and third stops to 4,000,000 (which the chart uses, it continues doubling the previous values). However, camera makers appear to be using the precise values above 10000 (higher numbers, also shown in the chart). For nominal half stops in the chart, I just continued doubling the previous half stop values.

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