Given two functions $f_1(m) = a^m$ and $f_2(m) = b^m$, how to design another function $f(m)=g(f_1(m), f_2(m))$ such that $f(m)$ is maximized at some finite value $m=m_o$ (with $m_o$ not equal to $0$ or $\infty$). Or prove there is no such function $g(\cdot,\cdot)$.
For instance, $g(x,y) = \frac{x}{y}$ does not meet the requirement, since if $a\le{}b$, $m_o=0$. If $a\ge{}b$, $m_o=\infty$.