Skill: Interpret logarithms as exponents

A logarithm tells you the exponent needed to produce a number:

$$\log_b(a)=c \quad \Longleftrightarrow \quad b^c=a$$

Here, $b$ is the positive base, $a$ is the positive number inside the logarithm, and $c$ is the exponent.

Example: Find $\log_2\left(\frac18\right)$.

Let

$$x=\log_2\left(\frac18\right).$$

Use the definition of a logarithm to rewrite this as an exponential equation:

$$2^x=\frac18.$$

Rewrite $\frac18$ as a power of $2$:

$$\frac18=\frac{1}{2^3}=2^{-3}.$$

Therefore,

$$2^x=2^{-3}.$$

Since the bases are the same, the exponents must be equal:

$$x=-3.$$

So,

$$\boxed{\log_2\left(\frac18\right)=-3}.$$

This means that the exponent required to produce $\frac18$ from base $2$ is $-3$, because

$$2^{-3}=\frac18.$$