A logarithm answers an exponent question. For $a>0$, $a\ne1$ and $x>0$,

$$\log_a x=y\qquad\Longleftrightarrow\qquad a^y=x.$$

For example,

$$\log_2 8=3$$

because $2^3=8$.

Logarithms as inverse functions

The logarithmic function reverses the exponential function:

$$\log_a(a^x)=x,$$

and

$$a^{\log_a x}=x\qquad (x>0).$$

Its domain is therefore the positive real numbers. The graph passes through $(1,0)$ because $\log_a1=0$.

Logarithm laws

Exponent laws become logarithm laws. For positive $x$ and $y$,

$$\log_a(xy)=\log_a x+\log_a y,$$

$$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y,$$

and

$$\log_a(x^r)=r\log_a x$$

whenever the expressions are defined.

Common and natural logarithms

Base $10$ logarithms are often written $\log x$. The logarithm with base $e$ is the natural logarithm,

$$\ln x=\log_e x.$$

Because logarithms and exponentials are inverses, applying one can turn an exponential equation into an ordinary algebraic equation. For example,

$$e^x=5\qquad\Longrightarrow\qquad x=\ln5.$$