# Reciprocal of Logarithm

## Theorem

Let $x, y \in \R_{> 0}$ be (strictly) positive real numbers.

Then:

$\dfrac 1 {\log_x y} = \log_y x$

## Proof

 $\displaystyle \log_x y \log_y x$ $=$ $\displaystyle \log_y y$ Change of Base of Logarithm $\displaystyle$ $=$ $\displaystyle 1$ $\displaystyle \leadsto \ \$ $\displaystyle \log_y x$ $=$ $\displaystyle \dfrac 1 {\log_x y}$

$\blacksquare$

## Examples

### Logarithm Base $10$ of $2$

The reciprocal of $\log_{10} 2$ is $\log_2 10$.