# Statistics Dictionary

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### 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 = bx. Then,

logb(y) = x   or   logb(bx) = 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 = 10log(x) and y = 10log(y)

Therefore,

xy = 10log(x) * 10log(y) = 10log(x) + log(y)

so

log(xy) = log(10log(x) * 10log(y)) = log(x) + log(y)

Also,

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