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 $\log_{2}\left(8\right) = 3$, 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 $\log\left(1000\right) = 3$. And $\ln\left(x\right)$ 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. $\log_{b}\left(1\right) = 0$, 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 $\log_{2}\left(0\right)$ and $\log_{2}\left(-4\right)$ are undefined. The input of a log must be positive.