Logs of Fractions

A negative power makes a fraction: ${2}^{-4} = \frac{1}{{2}^{4}}$, which is $\frac{1}{16}$. Read backwards, that means $\log_{2}\left(\frac{1}{16}\right) = -4$. When the input is 1 over a power of the base, the log is the negative of that exponent.

Here is why the answer must be negative. With base 2, positive powers give numbers above 1, and the power 0 gives exactly 1. To land below 1 you need a power below 0. So with any base above 1, an input between 0 and 1 has a negative log.

Decimals work the same way: $0.001$ is $\frac{1}{1000}$, which is ${10}^{-3}$, so $\log\left(0.001\right) = -3$. Keep the two ideas apart: the input of a log must be positive, but the answer can be negative.