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

\(\ds \log_x y \log_y x\)

\(=\)

\(\ds \log_y y\)

Change of Base of Logarithm

\(\ds \)

\(=\)

\(\ds 1\)

\(\ds \leadsto \ \ \)

\(\ds \log_y x\)

\(=\)

\(\ds \dfrac 1 {\log_x y}\)

$\blacksquare$

Also presented as

This result can also be seen presented as:

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

Examples

Logarithm Base $10$ of $2$

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