Can a logarithm have a function as a base?

Question

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

Answer 1

Can a logarithm have a function as a base ?

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

Answer 2

short answer: no.

I would add that if you ever felt the need to write $\log_{\sin x}(3x)$, you could simply write $(\sin x)^y=3x$ and solve for $y$. Of course there is no solution for all $x$ in this case. as a different example, you could have something like $(x^2)^k=x^6$, and clearly $k=3$. in a different world, you could write this as $\log_{x^2}x^6=k$, but this is just not standard notation.

I also would point out that logarithms were invented as a way to handle large numbers. In that sense, they were originally a matter of convenience, and it doesnt really make sense to use them for something like $\sin x$ since we have no problem writing regular functions down.