Complex problems can be calculated easily and accurately by means of logarithms. You can add logarithms to achieve multiplication, subtract them to achieve division, and divide them to derive square roots.

In photographic quality assurance, logarithms are used for the following:

Determining density

*Plotting characteristic curves

Determining contrast

Determining log H

*Reading the densitometer scale

A common logarithm (log _l0) is an exponent to a base number of 10. The base 10 is used because our numerical system is based on units of 10. This can be demonstrated easily by using scientific notation, or "powers of 10" For example, the logarithm of 100 is 2, because 102 equals 10 times 10, or 100. The logarithm of 1,000 is 3, because 103 equals 10 times 10 times 10, or 1,000. Table 2-1 shows how some common logarithms are computed. Notice the relationship between the exponent (superscript) and the common log.

Table 2-1.-Examples of Some Common Logarithms

The original number in table 2-1 is also called an antilog. Notice that a number greater than one is a positive log. Any number less than 1, but greater than zero, is a negative log.

Logs are also required between the numbers 1 and 10. Since the log of 1 is 0 and the log of 10 is l, the numbers 1 through 9 are decimals. (See Table 2-2)

Notice the relationships between numbers and their logs as follows:

*When numbers are multiplied, their logs are added. Example: 8 = 2 x 4. The sum of log 2 and log 4 equals log 8: 0.30 + 0.60 = 0.90.

When numbers are divided, their logs are subtracted. Example: 3 = 6 - 2. Log 3 is the difference between log 6 and log 2: 0.78 - 0.30 = 0.48.

The previous discussion is provided to give you a general idea on how logarithms are derived. It is not necessary for you to memorize logarithms, or refer to the log tables. All scientific calculators have a "log" key that converts numbers to logarithmic form. You should become familiar with the functions of your calculator before proceeding with the study of photographic quality assurance. For more information on using logarithms, refer to the chapter on logarithms in Mathematics, Volume 1, NAVEDTRA 10069.

One of the main uses of logarithms in photographic quality assurance is to take the numbers used to indicate exposure in characteristic curves and reduce them to a manageable form. For example, the

Table 2-2_-Common Logarithms Between 1 and 10

sensitometer in your imaging facility is set on an exposure time of 1/100 second and provides an illuminance of 80,000 lux (or meter-candles). The log exposure can be calculated easily as follows:

ExT=H

80,000 (lux) x 1/100 (set) = 8001ux seconds The log exposure = the log of 800 or 2.90

When you convert exposure to logarithmic form, both density and exposure are on the same scale. A characteristic curve indicates how exposure and processing differences affect photographic emulsions by comparing density and the log of exposure.

To describe sensitometry, you must become acquainted with several new terms and formulas. As a starting point, you should become familiar with the terms transmission, opacity, and density, or T, 0, and D.

TRANSMISSION

Most photographic material, even clear film, does not transmit all of the incident light that is relevant to it. Transmission is a measure of the light-passing ability of a film or other medium. The transmission of a processed film refers to the fraction, or percentage, of incident light that passes through the film.

In a formula, transmission is represented by a capital letter T. The formula for determining transmission is as follows:

T = Amount of transmitted light Amount of incident light

The result is never more than 1/1, or 1.00. For example, when 10 meter-candles (mc) of light are incident (or falling) to a film and 5 me is passing through it, the transmission is as follows: T = 5/10 or T = 0.50, or 50 percent. When 2 me is transmitted, the formula reads T = 2/10 or 0.20, or T = 20 percent. OPACITY

Opacity is the ability of a medium to absorb light. The two terms, transmission and opacity, are directly opposite in meaning. Opacity is indicated in a formula by the capital letter 0 and is the reciprocal of the transmission (T). The formula for opacity is given as follows:

You can see that opacity is the transmission formula inverted. Again, when a material has 10 me of light falling on it and 5 me is being transmitted, you can determine the opacity by the formula O = 10/5, or 2. Putting it a different way, opacity is the reciprocal of transmission, or O = 1/0.50 = 2. Or, when 2 me is being transmitted, the formula is O = 1/0.20, or O = 10/2; in either case, the opacity is 5.

DENSITY

Density is indicated by the capital letter D and is another way of expressing opacity or the light-stopping ability of a medium. Density is nothing more than the common logarithm of opacity. For example, when opacity is 2.0, the density is log 2 = 0.30.

In sensitometry, density is the term with which you are most concerned. However, density cannot be disassociated from transmission and opacity, because they all are dependent and directly related to each other. When the value of any one of these factors is known, you can calculate the others. When you know the transmission of a film, you can easily determine the density. Or conversely, you can measure the density and then determine the amount of transmission. While charts are available to provide some conversions directly, you should be capable of determining any of those figures.