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.