Does there exist a function which converts exponentiation into addition? A useful property of the logarithm is that it can "convert" multiplication into addition, as in
$\ln(a)+\ln(b)=\ln(ab) \text{ for all } a, b \in \mathbb{R}^+$
Does there exist a function $f$, which holds a similar property for exponentiation?
$f(a)+f(b)=f(a^b) \text{ for all } a, b \in \mathbb{R}^+$
If so, are there any closed-form expressions for such a function?

As Stinking Bishop mentions in the comments, if $f(a)+f(b)=f(a^b)$, then it follows (by interchanging $a$ and $b$) that $f(b)+f(a)=f(b^a)$. Hence, $f(a^b)=f(b^a)$ for all $a,b\in\mathbb R^+$. Setting $b=1$, we see that $f(a)=f(1)$ for all $a\in\mathbb R^+$. Now $f(1)+f(1)=f(1^1)$, so $f(1)=0$. Hence, the only function $\mathbb R^+\to\mathbb R$ with the desired property is the zero function.
