Is it possible to prove $\exists x. a^x = b$ without logarithms?

Question

Let $a>0$ and $b>0$ be real numbers such that $a \ne 1$. Then can

$\exists x. a^x = b$

be proven without using logarithms?

Answer 1

Yes: using the fact that the function is continuous, that it diverges on one side and that it tends to zero on the other.