Is $\log(e^{x^a + x^b})$ a convex function?

Question

Where $a$ and $b$ can be any real numbers, including fractional or negative. But $x$ is positive.

Answer 1

Hint: $\log(e^{x^a+x^b})$ simplifies to $x^a+x^b$, which is twice differentiable, so it's convex iff its second derivative is nonnegative over its domain (in your example the domain is $x>0$). This should lead to a counterexample. For simplicity try $a=b$.