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I have a simple question regarding an identity of logarithms and I was hoping someone could provide some insights or guidance.

On the Wikipedia website, they have the following identity for a base ${ xy}$:

${\displaystyle {\frac {1}{{\frac {1}{\log _{x}(a)}}+{\frac {1}{\log _{y}(a)}}}}=\log _{xy}(a)}$

I was wondering what would happen if we allowed the base to be the product of more than two integers.

Thanks in advance!

AmB
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1 Answers1

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This is just repeated use of $\,\log_x(a)=\frac 1{\log_a(x)}$ and the sum to product formula.

$$\log_{xyz}(a)=\frac 1{\log_a(xyz)}=\frac 1{\log_a(x)+\log_a(y)+\log_a(z)}=\frac 1{\frac 1{\log_x(a)}+\frac 1{\log_y(a)}+\frac 1{\log_z(a)}}$$

And you can extend to any number of terms.

zwim
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