Rewriting basic functions in polar form

Question

I've been exploring how to rewrite common parent functions ($x^2, \sqrt x$,...) in polar form.

Is it possible to rewrite natural log or trig functions in polar form as a function of $\theta$?

For example, I tried to rewrite $ln(x)$ using simple substitution:

$y=ln(x)$

$r\cdot sin(\theta)=ln(r\cdot cos(\theta))$

And I assume after this I am unable to separate the r's. I find the same issue with sin, cosine, exponential...

Also, if it is impossible, is there an explanation why aside from the fact that the variables can't separate?

Answer 1

One of the problems comes from the fact that for common functions such a dependence $r=r(\theta)$ will be multivalued (logarithm is concave, so you are lucky, at most two values for a given $\theta\in(-\pi/2,\pi/2)$; trigonometric functions - not even close, the numer of values taken by $r(\theta)$ depends heavily on $\theta$).

Another problem is that such dependence will almost never be expressed in terms of common functions.