# 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.)