Reciprocal of Logarithm

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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 $\log_2 10$.