calculus – (algebraically) prove $a^x$ is an increasing function

How to prove $a^x$ (for a>1) is an increasing function, independently of $int_1 ^x (frac{1}{x})$ or Ln(x). Is there a way to algebraically prove $(x_1>x_2 longrightarrow a^{x_1}>a^{x_2})$ with as little reference possible to Calculus? $$$$
I understand we technically define $a^x$ through Ln(x), but is there a different algebraic perspective that we can give to $a^x$? At least for the rational numbers (since we need the Ln(x)-based definition for defining irrational powers.)