T.O. 33B-1-16-306.4.1.5.1Logarithms provide a convenient method of multiplying and dividing. To multiply two numbers, you take thelogarithms of both numbers and add them to get the logarithm of the product. To obtain the product, you then take theantilogarithm of this sum. The antilog is the inverse function of the log. In other words, the antilog of x is equal to10^{x}. Table 6-11 lists the antilogs for mantissas of 0.000 to 0.999. The value of x is then obtained by properly placingthe decimal point according to the characteristic of the sum of the logarithms.Example: Multiply 20 times 8 using logarithms.(1) Take the log of 20: log 20 = 1.3010(2) Take the log of 8: log 8 = 0.9031(3) Take the sum of these logarithms: 1.301 + 0.9031 = 2.2041(4): Take the antilog of the sum:The antilog of 0.2041 = 1600The characteristic of 2 indicates a number between 100 and 999.Therefore the answer is 160.This of course is the answer that we expected. In this example, the regular mathematical calculation is simpler.However, with very large numbers, the use of logarithms significantly simplifies calculation.6.4.1.5.2Division is accomplished by taking the difference between the logs of the two numbers.Example: 6/73 = antilog (log 6 - log 73).6.4.1.5.3In radiography, logarithms find particular use in the preparation of exposure charts and in film characteristic curveswhich plot film density against relative exposure as explained in paragraphs 6.4.1.6. Logarithms to the base 10 may beconverted to natural logarithms by the equation ln x = 2.3 log x.6.4.1.6 CharacteristicCurve.The characteristic curve is the response of a type of film to radiation of a particular energy. It is obtained by plottingthe film-image density against the logarithm of relative exposure. Since density is a logarithm, log-log scales are usedfor the plot. Log-log scales not only make interpretation of the graph easier, but also the all important values of relativeexposure can be derived easily by subtracting one logarithm value from another. Study of Figure 6-14, shows that atlow exposures, a large change in exposure is needed to produce a significant change in density. As the relativeexposure increases, the film emulsion becomes more sensitive and the same exposure change produces a greater densitydifference. The gradient (slope) of the curve in Figure 6-14 increases with increasing exposure. At very high valuesthe gradient may start to decrease; that is, the film again becomes less sensitive. However, this effect, althoughcommon with medical film, is not often encountered in industrial radiography. The term used to refer to the gradientof the characteristic curve is "film contrast."