Is $\ a^b+b^a\ $ transcendental if $\ a^b\ $ and $\ b^a\ $ are?

Question

Suppose, $\ a\ $ and $\ b\ $ are positive real numbers such that $\ a^b\ $ and $\ b^a\ $ are both transcendental.

Can we conclude that $$a^b+b^a$$ is transcendental ?

I tried to use $$b^a=(a^b)^{a/b}\cdot (\frac{b}{a})^a$$ but this led to nowhere.

Answer 1

Let $f(x)=(\frac{1}{\ln(x)})^x$. Then $f(2) \gt 4-e \gt f(3)$, so by continuity there is an $a$ such that $f(a)=4-e$. For this $a$, if we put $b=\frac{1}{\ln(a)}$ we have $a^b=e$ and $b^a=4-e$ yielding a counterexample.