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By Ed Buffaloe When making test exposures for prints, most people just do a series of 3 or 5 second exposures across the paper, as recommended by Fred Picker, among others. This linear series of exposures does not take into account the fact that exposure progressions are inherently geometric in nature. To grasp this easily, imagine the difference between adding 3 seconds to a 3 second exposure and adding 3 seconds to a 30 second exposure. In the first case the exposure is doubled (i.e., one full stop more exposure), whereas in the second case the increase is almost negligible (i.e., about 1/8 stop more exposure). Since 1/4 stop is about the minimum exposure change the unaided eye can differentiate, it would be most convenient if we could make our test strips in exact 1/4 stop increments. But how can this be done? The key lies in the logarithms of base 2. You don't need to understand logarithms to use this exposure technique, so if you wish you can skip to the bottom of the article and simply utilize the exposure progressions and multipliers outlined there. First it is necessary to understand the logarithmic nature of full stops. Photographic exposure progressions are based on numbers that are factors of 2, such as 1, 2, 4, 8, 16, 32, 64, etc. 2^{0} = 1 Log_{2} of 1 = 0 Any exposure multiplied by 1 (log_{2 }of 0) gives plus 0 stops more exposure. So when we say X stops more exposure, the number we multiply by is the base 2 log of X. Most of us just understand intuitively that these numbers are 2, 4, 8, 16, etc., without knowing they are the logarithms of 2. Now for the 1/4 stop increments. Let us fill in the blanks. 2^{0} = 1.00 Log_{2} of 1.00 = 0.00 To give +1/4 stop, multiply by 1.19. To give 1/4 stop, divide by 1.19. Here are two handy charts: Multipliers: 3/4 stop: multiply by .59
1/4 stop Exposures, beginning with 5 seconds:
Copyright 1999 by Ed Buffaloe.
Logarithms
Start with: b^{x }= y. b = base If the base and exponent are given we compute a power. The logarithm of a number y with respect to a base b is the exponent to which b must be raised to obtain y. x = log _{b }y or b^{x} = y Examples: 10^{2} = 100 log_{10 }100 = 2 The most important logarithms in photography are those of base b = 2. Logarithms of base b = 10 are known as common logarithms. Logarithms of base e = 2.71828… are known as natural logarithms. Ed Buffaloe has written articles articles for PostFactory Photography magazine, and is working on a howto darkroom web site. He can be reached at edbuffaloe@earthlink.net.

