22

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!

abiessu
  • 8,115
Spencer Fleming
  • 331
  • 4
    Remember that logarithms should be taken respect to positive bases. – Pedro Aug 14 '15 at 03:54
  • 4
    That's true, otherwise the answer is a complex number, correct? Would a function that must be positive and !== 1 work, for example $log_{x^2 + 2}(3x)$? – Spencer Fleming Aug 14 '15 at 04:02

2 Answers2

42

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.

Lucian
  • 48,334
  • 2
  • 83
  • 154
  • 6
    A short version of the essential point of the answer (to the intended but unasked question): The base of a logarithm should be a number. For every $x$, $\sin x$ is a number. – Andreas Blass Aug 14 '15 at 04:58
  • Those are some good points, yes that's what I meant, and thanks for the advice. Out of curiosity if not function, what should sin(x) be called? Can logarithms with results of functions as their base be derived using the standard $\frac{1}{x\ln(a)}$ or would the chain rule apply since it is differentiable? – Spencer Fleming Aug 14 '15 at 05:05
  • @user31157 As an exercise, try computing the derivative of $log(3x)/log(sin(x))$ and see if it works in the example case given. I'm rusty, so I could have made mistakes, but I find an answer nowhere near $1/x ln(a)$ nor near what you'd get with the chain rule, because there's a division in the midst. In case of a numeric constant base, that's no problem, but with a function, you have to pay attention. – Cort Ammon Aug 14 '15 at 07:06
  • 1
    The base doesn't have to be a number, does it? Like, considering that $f^2 \equiv f \circ f$, I wouldn't mind saying $\log_f f^n = n$ if $f$ is a function... – user541686 Aug 14 '15 at 11:16
  • 3
    @Mehrdad I would say it's some kind of notation abuse: understandable, reasonable but formally wrong. – seldon Aug 14 '15 at 11:24
  • @mattecapu: What if $f$ is a matrix? Do you still consider it wrong? – user541686 Aug 14 '15 at 11:27
  • 2
    @Mehrdad: There are a number of ideas very closely to functions that (in standard mathematical language) can be used like that. But if you truly mean to talk about functions, then that is grammatically incorrect. And part of the reason (or maybe the entire reason) you see this abuse is that these other ideas aren't really taught at an introductory level, but people use functions in the way these other ideas would be used. (these ideas include things like (dependent) variables, scalar fields, generalized elements, and various sorts of algebras) –  Aug 14 '15 at 11:56
  • @Mehrdad as far as I know, the most common definition of the logarithmic function uses real numbers for the base and the argument, so any other mathematical entity you substitute to those numbers needs an explicit definition that bridges the gap between the base definition and your new use of the log function. This to be pedantically correct, but in turn, that's what doing mathematics means ;) – seldon Aug 14 '15 at 12:27
  • @Mehrdad: I thought of the exact same thing you did when writing my answer, but since that was not the definition of a logarithm that the user had in mind, I saw no reason to over-complicate the post. – Lucian Aug 14 '15 at 12:39
  • Vacuously, it is a ridiculous equation? – BCLC Aug 15 '15 at 05:15
3

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.

pancini
  • 19,216
  • 1
    $(\sin x)^y=3x$ has a (trivial) solution set at $x=0$, but maybe you meant something else by "there is no solution..." – abiessu Aug 14 '15 at 04:06
  • agree; i should have written "for all x" – pancini Aug 14 '15 at 04:08
  • 1
    This makes sense, especially considering that logarithms must have a positive base to make sense without adding complex numbers, which I forgot until Pedro pointed it out. Out of curiosity: do you suppose a log like your $log_{x^2}x^6$ would derive like other logs, with $\frac{1}{xln(a)}$ or would it require a chain rule since it would be differentiable, or simply be impossible? – Spencer Fleming Aug 14 '15 at 04:16
  • i suppose it would vary but in the case the derivative would be zero since it is constant – pancini Aug 14 '15 at 04:18
  • Right, I forgot about that, thanks. The reason I first went hunting for logs with function bases in the first place was to find the dirivitive $log_{log_{sin^{-1}(sin(x))}12x}3x^2$ and I wasn't sure it was even possible. I suppose since it would cross x=1 or x<0 quite often it wouldn't really work without complex numbers even if it was the correct notation. Thanks for all the help! – Spencer Fleming Aug 14 '15 at 04:32
  • I thought logarithms were primarily to do multiplication and division as addition (for example with the help of my book with tables of natural and common logarithms). Also the logarithm made slide rules possible when the rule was marked with a logarithmic scale. – sesquized Aug 14 '15 at 16:23
  • of course they have many applications but napier invented them originally as a way to handle large numbers – pancini Aug 14 '15 at 19:30