Isomorphism with operation $a^{log b}$

Question

On a homework assignment on a course on group theory we tot asked

Prove that $G_{, \cdot} \cong \mathbb{R}^{+}$ where $G=\mathbb{R}_{>1}$ with the operation $a \cdot b=a^{\log b}$

So I figured the challenge was finding a bijection $f:\mathbb{R} \rightarrow \mathbb{R}_{>1}$ with the constraint $f(a+b)=f(a)^{\log f(b)}$. I tried $f(x)=e^{x}+1$ bit that didn't work out. Am I overlooking something trivial, or should I focus on proving the existence of the isomorphism instead of finding a function?

Answer 1

Hint: Try to rewrite the operation $a\cdot b$ in a more symmetric form, where $a$ and $b$ have similar roles. Does this form suggest an $f$ to you?