beninar6u

2022-09-08

Can a logarithm have a function as a base?
For example is ${\mathrm{log}}_{\mathrm{sin}\left(x\right)}\left(3x\right)$ a ridiculous equation?
I couldn't find an example on any page about logarithms that used a function on a base, but it seems that for an equation like $\mathrm{sin}\left(x{\right)}^{12x}$, the log's base would have to be the sine function. Thank you for the advice!

Jasmin Hoffman

"Can a logarithm have a function as a base ? "
Of course not ! But, then again, $\mathrm{sin}\left(x\right)$ is not a “function” ! Rather, it is the value of a function — in this case, the sine function — evaluated at point x. These are two different concepts ! Related, to be sure, but different nonetheless.
"Is ${\mathrm{log}}_{\mathrm{sin}\left(x\right)}\left(3x\right)$ a ridiculous equation ?"
Of course not ! In order for an expression to be a “ridiculous equation”, it must be an “equation” first. But I see no equality signs there — do you ?
Now that I'm done answering the questions you did ask, allow me to answer the one you never actually asked, but probably meant to all along: Yes, the mathematical expression ${\mathrm{log}}_{\mathrm{sin}x}\left(3x\right)$$=\frac{\mathrm{log}\left(3x\right)}{\mathrm{log}\mathrm{sin}x}$ makes perfect sense, assuming x lies inside positive intervals for which $\mathrm{sin}x$ is also positive.

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