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, for example EV and EV Chart for page evchart.html are shown, and then also the charts of camera nominal and precise setting values for page fstop2.html. Several of the calculators here use these same basics.
If these first basics are not clearly obvious, a longer version follows below.
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 and squared (because f/stop numbers increase exposure running in the opposite direction).
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 = log_{10}(n²) / log_{10}(2) = -4 EV difference (log of fractions are negative)
Or n = 11.314/2.828 = 4 f/2.8 compared to f/11
EV = log_{10}(n²) / log_{10}(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 = log_{10}(n) / Log_{10}(2) = 4 EV difference.
Shutter speed and ISO numbers and ratios are Not reversed or squared, only f/stops
The EV formula, see Wikipedia EV
N is f/stop Number, t is duration Time of shutter speed — the camera settings.
Log₂ creates EV as an exponent of 2:
The reflected light meter Exposure formula:
See Wikipedia light meter calibration
where L is the scene luminance, S is the ISO sensitivity, and K is a constant typically 12.5.
The only purpose here is to just show there is a formula where ISO combines with the scene luminance to match a proper exposure with the camera settings expressed as EV. We don't have to know it, but it is how light meters work. Meter results show one of the setting combinations, or some meters can show the EV number too.
The second formula 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 meter formula, and the main math thing computing does need to know is:
N is f/stop Number and t is shutter speed (duration Time). Note f/stop NUMBERS are squared. The log₂ method 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, 2^{10} = 1024.
EV is the exponent of 2 that computes the value, again meaning 2^{10} = 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 most 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 2^{1} 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 2^{0} = 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 numbers not necessarily actually existing due to the approximation). Back then, the binary numbering sequence of 1, 2, 4, 8, 16, 32, 64, etc. was not as mainstream as today. Modern cameras 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). No claim is made here about actual camera accuracy, but these are instead the exact PRECISE GOALS of design, 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 a circle is Pi × r². Doubled area is 2 (Pi × r²) = Pi × (√2 × r)². So 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 to compute the exposure value. It 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: 2^{3} = 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 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}), same math.
f/stop Nominal | 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 | f/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} (which steps are the powers of 2).
Example: 2^{-7} = 1/128 second
S.S. Nominal | 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 “smoother” 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 1.4^{5} = 5.38, but 1.4142^{5} = 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.)
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.
EV = log₂ (fstop² / shutter time)
Points about photographic EV ratios:
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 = log_{10}(n²) / log_{10}(2) = -4 EV difference (log of fractions are negative)
Or n = 11.314/2.828 = 4 f/2.8 compared to f/11
EV = log_{10}(n²) / log_{10}(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 = log_{10}(n) / Log_{10}(2) = 4 EV difference.
Shutter speed and ISO numbers and ratios are Not reversed or squared, only f/stops.
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 = log_{10}(n²) / log_{10}(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 = log_{10}(n²) / log_{10}(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.
EV_{S} = EV_{100} + log₂ (S / 100) where S (speed) is the new ISO value (converted FROM EV_{100}, 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 EV_{800} at ISO 800:
EV_{800} converted to a New ISO S = EV_{800} + 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.
EV_{100} 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. Only EV at the same ISO can be numerically compared for exposure difference or scene brightness, and that convention is ISO 100.
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 EV_{100} from EV_{S}, which common convention and I name to be used for Light Value:
EV_{100} = 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) = EV_{100} = 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 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've 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). Examples:
log_{10}(8) = 0.90309 the logarithm
10^{0.90309} = 8 base^{exponent} is the value again, called antilog of the logarithm
In base 10, 10^{1} = 10, 10^{2} = 100, 10^{3} = 1000 (each step 10x value)
log_{2}(8) = 3 the logarithm ( log_{2}(X) = log_{10}(X) / log_{10}(2) )
2^{3} = 8 base^{exponent} is the value again, called antilog of the logarithm
In base 2, 2^{1} = 2, 2^{2} = 4, 2^{3} = 8 (each step 2x value, important in cameras)
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. Relative to the other light, the log_{2}(8) = 3 means in 2x steps, our light is doubled 3 times, or 8 times more light or power, because 2^{3} = 8. 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 = log_{2}(intensity ratio).
From the 1600s until the later parts of the 1900s (before computers, the time basically from John Napier and Newton to 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.
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 log_{10} is 0.4971495)
If the radius of a circle is 122.125 (and that log_{10} is 2.0868046)
then the area of the circle is Pi r² = 3.14159 × 122.125² = 46855.3
Or with logs, the log of the area is 0.4971495 + (2.0868046 + 2.0868046) = 4.6707587
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. :) Adding or subtracting many digits is much easier than multiplying or dividing them.
Log and exponent are just reverse operations, opposites. We know that addition/subtraction and multiply/divide are also reverse or opposite or inverse operations. One can "undo" the other, so to speak. Log and exponent are also the similar reversed relationship. So 2^{3} = 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 (powers of 2, so to speak). So regarding photographic exposure calculations, our exposure in powers of two are just log₂ values. And the log₂ of ratio of two values is also the EV difference. What we need to know now is how to compute log₂.
EV = log₂ (fstop² / shutter time) (Using precise values of settings)
To compute log₂, then log₂(X) is log_{10}(X) / log_{10}(2)
Or also, log₂ is log_{e}(X) / log_{e}(2)
(can use base 10 or base e, doesn't matter so long as consistent. See next Warning)
which are fine approximations, but... you should know that programming in Excel and Javascript and C and Python and other programming languages use Natural log base e instead of Common log base 10. Doesn't matter, results are the same if consistently same base. However these numerical constants (of 0.3 and 3.3) will give the wrong answer (only due to the constants, because log_{10}(2) = 0.30103 and log_{e}(2) = 0.693147). Using 0.3 for log_{e}(2) is simply wrong.
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 Common log_{10} processing. To compute log₂(X) is log_{10}(X) / log_{10}(2).
On a Texas Instruments calculator, these key stroke sequences are (the value of X) log ÷ 2 log =
Examples of log bases:
log_{10}(2) = 0.30103
log₂(2) = 1
log_{e}(2) = ln(2) = 0.69315 (JavaScript returns this constant with Math.LN2)
Any log base computes correctly if you are consistent, but NOT if you specify a wrong constant for the log base used.
Handheld scientific calculators (and the Windows Calc program) likely have a key called lnx or ln which is Natural log_{e}, and another key called log which is Common log_{10} to simply do that conversion (it divides by log_{e}(10)). 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. 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 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 2^{6} = 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):
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 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 StopNumber 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 2, or like 1/2 computing 0.5, are fully precise values, 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).
The value 1 (as f/1 or 1 second) is Stop Number 0 (because any number to power 0 is 1).
Shutter speed is computed as base 2 to exponent Stop Number. The nominals in the text can appear their way.
F/stop is computed as base √2 = 1.4142 to exponent Stop Number.
Fractional stops are computed with exponent (Stop Number + fraction).
Example: f/5.6 + 2/3 EV = (√2)^{(5 + 0.666667)} = f/7.12719
Nominal values | |||||||||||
shutter speed | 1/30 | 1/15 | 1/8 | 1/4 | 1/2 | 1 | 2 | 4 | 8 | 15 | 30 |
---|---|---|---|---|---|---|---|---|---|---|---|
f/stop | 0.5 | f/0.7 | f/1 | f/1.4 | f/2 | f/2.8 | f/4 | f/5.6 | |||
Precise computed values of goal | |||||||||||
Stop Number | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
2^{Stop Number} | 1/32 | 1/16 | 1/8 | 1/4 | 1/2 | 1 | 2 | 4 | 8 | 16 | 32 |
(√2)^{Stop Number} | 0.5 | 0.707 | 1 | 1.414 | 2 | 2.828 | 4 | 5.657 |
EV calculations require logarithm math detailed above.
ISO requires a modified method (below).
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 | |
Precise | Nominal | ||
-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
2^{3} is 8 seconds 2^{-3} is 1/8 or 0.125 seconds | |||
(√2)^{StopNum} is f/stop number
1.414^{3} 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. Except 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 (0^{0} 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 2^{0} 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.
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.
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:
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 Exponent is Stop Num | |
---|---|
2^{N} | Duration |
2^{5} | 32 sec. |
2^{4} | 16 sec. |
2^{3} | 8 sec. |
2^{2} | 4 sec. |
2^{1} | 2 sec. |
2^{0} | 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 2^{0} = 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 = 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).
Aperture Exponent is Stop Num | |
---|---|
(√2)^{N} | f/Number |
1.414^{0} | 1 |
1.414^{1} | 1.414 |
1.414^{2} | 2 |
1.414^{3} | 2.828 |
1.414^{4} | 4 |
1.414^{5} | 5.657 |
1.414^{6} | 8 |
1.414^{7} | 11.314 |
1.414^{8} | 16 |
1.414^{9} | 22.627 |
1.414^{10} | 32 |
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 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(2^{3}) = 2.828 (seen sometimes).
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
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 2^{0}). At the time, the European DIN rating also used 2^{0} as a base. Using the standard Stop Number rule that 2^{0} 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 2^{5} = 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 the third stops have been called. Nominals shown in this second chart are from the first ISO chart above, probably one reason it changed. Rounding sometimes matches conventional nominal, and sometimes does not. Generally for shutter speed and ISO, one stop greater than any nominal third should be 2x the numerical value (2x number is two stops for f/stop).
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.
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. That 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.
2^{5} 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.