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!
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.