beninar6u

2022-09-08

Can a logarithm have a function as a base?

For example is ${\mathrm{log}}_{\mathrm{sin}(x)}(3x)$ 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 $\mathrm{sin}(x{)}^{12x}$, the log's base would have to be the sine function. Thank you for the advice!

For example is ${\mathrm{log}}_{\mathrm{sin}(x)}(3x)$ 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 $\mathrm{sin}(x{)}^{12x}$, the log's base would have to be the sine function. Thank you for the advice!

Jasmin Hoffman

Beginner2022-09-09Added 6 answers

"Can a logarithm have a function as a base ? "

Of course not ! But, then again, $\mathrm{sin}(x)$ 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 ${\mathrm{log}}_{\mathrm{sin}(x)}(3x)$ 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}(3x)$$={\displaystyle \frac{\mathrm{log}(3x)}{\mathrm{log}\mathrm{sin}x}}$ makes perfect sense, assuming x lies inside positive intervals for which $\mathrm{sin}x$ is also positive.

Of course not ! But, then again, $\mathrm{sin}(x)$ 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 ${\mathrm{log}}_{\mathrm{sin}(x)}(3x)$ 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}(3x)$$={\displaystyle \frac{\mathrm{log}(3x)}{\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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