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
\(\ds \log_x y \log_y x\) | \(=\) | \(\ds \log_y y\) | 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 $\log_2 10$.