Statistics and Probability Dictionary

Logarithm

A logarithm is an exponent. The logarithm of a number to the base b is the power to which b must be raised to produce produce the number. Thus, suppose y = b^x. Then,

log_b(y) = x   or   log_b(b^x) = x

The logarithm of a number to the base 10 is called the common logarithm. By convention, log(N) refers to log₁₀(N). Logarithms can be used to express any number greater than zero. Here are some examples.

log₁₀(1000) = 3;    log₂(32) = 5;    log₅(1) = 0;    log(0.01) = -2;    log(1.49) = 0.173

Since logarithms are exponents, they follow the same mathematical rules as exponents. We illustrate these rules below.

Let x = 10^log(x) and y = 10^log(y)

Therefore,

xy = 10^log(x) * 10^log(y) = 10^{log(x) + log(y)}    so    log(xy) = log(10^log(x) * 10^log(y)) = log(x) + log(y)

Also,

x^r = [10^log(x)]^r    so    log(x^r) = r * log(x)