How to find period of a function?

Question

Let’s say we have $2$ functions - $f$ and $g$.

I know that the period of the function $f+g$ or $f-g$ is the L.C.M. of the periods of $f$ and $g$.

What about the period of functions of the form $fg$ and $f/g$?

Answer 1

The period of $fg$ is also the LCM of the individual periods. To see this, just take the logarithm $$ \ln(f(x)g(x))=\ln(f(x))+\ln(g(x)). $$ We know that $\ln$ is a one to one function, so the period of $\ln(f(x))$ is the same as the period of $f(x)$. Then you can apply your result of $f+g$ and $f-g$.

Strictly speaking, $\ln$ only apply to positive functions, but I will omit such technical details here.

NB The LCM rule is not necessarily true: there are many counter-examples already posted on MSE.