What a Log Is

You know that 2^3 = 8. A logarithm asks that fact backwards: which power of 2 gives 8? The answer is 3, and we write log_2(8) = 3, read "log base 2 of 8 is 3". A log answers one question: which power?

In general, log_b(y) = x means exactly the same as b^x = y. The small number b is the base, the number you raise. It is always positive and never 1. Two bases have names of their own. A log with no base written is base 10, so log(1000) = 3. And ln(x) is the log with base e, a number near 2.718 that you will meet later in this section.

Two values hold for every base. log_b(1) = 0, because any base to the power 0 is 1. log_b(b) = 1, because any base to the power 1 is itself. And some inputs have no log at all: a positive base raised to any power gives a positive number, never 0 and never a negative. So log_2(0) and log_2(-4) are undefined. The input of a log must be positive.