Can any expression be written in terms of the natural exponent?

Question

Let's start off with a simple function say $y = x$. Can it be written in terms of the natural logarithm? If so, are there any functions that cannot?

Answer 1

For any $x$ we have that

$$y=x \iff y=\ln e^x$$

and more in general

$$y=f(x) \iff y=\ln e^{f(x)}$$

Otherwise if we are interested in a $\log-\log$ identity

$$y=f(x) \implies \ln y = \ln (f(x))$$

is true only for $f(x)>0$.

Answer 2

There are plenty of functions which cannot. For example, consider Ackermann's Function. It can be rigorously shown that $A(x)$ is not expressible in any way save recursively. You could add an $\ln$ to the definition, but it would still not be expressible in closed form.