# Detection Theory: A User's Guide

By Neil A. Macmillan; C. Douglas Creelman | Go to book overview

Appendix 2
Logarithms and Exponentials
All logarithms are defined with regard to a constant called a base. The logarithm, or log, of a number is the power to which the base must be raised to obtain the number. Thus, if the base is 10 (as in “common” logarithms), log(l) = 0, log(10) = 2, log (1000) = 3, and so on. As this example illustrates, the logarithm of a number increases monotonically with the number itself, but not nearly as quickly.Natural logarithms, the only kind appearing in this book, have abase of e = 2.718281828…. (The ellipsis indicates approximation: e cannot be exactly expressed as a fraction or repeating decimal. ) Some of the sensitivity and bias measures in detection theory are defined as the natural logarithm of other measures. Because the logarithm is a monotonic transformation, two measures related by it are equivalent (i.e., have the same isosensitivity or isobias curves).The easiest way to compute a logarithm is with a hand calculator. On most intermediate (statistical or scientific) calculators, the button relevant to natural logarithms is labeled “ln” (“log” being reserved for common logarithms). If you enter the number 10, and then In, the calculator should display 2.302585.To calculate measures defined using logs, no deeper knowledge is needed. To follow derivations, however, some other facts are useful:
 1 Multiplication of numbers corresponds to addition of their logarithms: ln(xy) = ln(x) + ln(y). 2 Division of numbers corresponds to subtraction of their logarithms: ln(x/y) = ln(x) − ln(y). 3 Raising a number to a constant power corresponds to multiplying its logarithm by that constant: In(xa) = aln(x).

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