What a Logarithm Means

In $2^3=8$, 2 is the base and 3 is the exponent, also called the index or power. If the exponent is missing, a logarithm names it.

Read $\log_2 8$ as “log of 8 to base 2”: which power of 2 gives 8? The double arrow means the two statements say the same thing.

Powers need not be positive integers. A zero power gives 1; a negative power gives a reciprocal; a fractional power gives a root.

So $\log_2 1=0$, $\log_2(1/8)=-3$ and $\log_9 3=1/2$. A negative logarithm is allowed: it is the exponent, not a negative input.

For real logarithms, use $a>0$, $a\ne1$ and $q>0$. A positive base has positive powers, so no real exponent produces zero or a negative result.

Base 1 cannot identify one exponent: every power of 1 is 1. Negative bases do not give real values for every real exponent; for example, $(-2)^{1/2}$ is not real. They are not real logarithm bases.

A base between 0 and 1 is allowed: $(1/2)^3=1/8$, so $\log_{1/2}(1/8)=3$.