RE: How to translate log density readings to percent?

My apologies if this appears twice -- it is AWOL since this morning.

The task is to map a logarithmic scale to a linear scale. First, note that
the log scale is ratiometric and the percent scale is absolute. Therefore,
there is no finite log value for 100%. Zero transmission (100% density)
would be represented as log (1/0), or log density infinity.

Log density ("log D") is computed as log (reference reading/test reading).
Read this as, "the logarithm of (the reference reading divided by the test
reading))" or "the logarithm of (the amount of light with no test material
divided by the amount of light with the test material)." The logarithms
here are common logarithms (base 10).

Recall that the logarithm of a number is just the exponent to which the
base must be raised to generate the number. The common logarithm of 100 is
2, because 10 raised to the power 2 is 100. Most people are comfortable
with whole-number exponents (10 ^1 = 10, 10 ^2 = 100, 10 ^3 = 1000, etc.),
but not so comfortable with fractional powers (10 ^1.76, for example).

So, let's say that our light detector is linear and gives a reading of 100
with no test material present (this is arbitrary). A perfectly clear test
sample will also give a reading of 100, so its log density is log
(100/100), or log (1), which is zero [any number to the zero power is 1,
so the log of 1 is zero].

If the test material transmits only half of the light, the detector reads
50 and its log density is log (100/50), or log (2), which is 0.3 (actually,
it is 0.301029996...). If the test material transmits 1% (1/100) of the
light, the detector reads 1 and the log density is log (100/1), or 2. Each
factor of 10 adds 1 to the log, so a test sample that transmits 1/1000 of
the light [detector reads 0.1] has a log density of 3 (because 10 to the
power 3 is 1000) and a sample that transmits only 1/1,000,000 [one
millionth] of the light [detector reads 0.0001] has a log density of 6.
Now, for practical purposes in photography, that test sample passes "no
light." Yet the log density scale marches steadily on -- log density 9
denotes a test sample that transmits one billionth of the light, and so on.

So, to turn log density into percent density, use this formula:

     % D = (100/(10 ^(log D)))

where "^" indicates to raise 10 to the (log D) power.

Thus, a log D of 0.3 is (100/(10 ^0.3))
= 100/2 = 50%

A log D of 2 is (100/(10 ^2)) = 100/100 = 1%

A log D of 0.72 is (100/(10 ^0.72)) = 100/5.25 = 19.05%

Reflected density works the same, only the reference reading is taken from
a perfectly reflective diffuse surface. A log D (reflectance) of 1.9
(pretty good DMax for a print) has a % D (reflectance) of

100/(10 ^1.9)) = 100/79.4 = 1.26%

Best regards,

etienne
Received on Tue Sep 7 17:30:47 2004