Let $a>1$, and let $r$ and $s$ be any rational numbers. Show that $r<s \iff a^r<a^s$

Question

I'm currently trying to prove the following:

Let $a>1$, and let $r$ and $s$ be any rational numbers. Show that $$r<s \iff a^r<a^s$$

I have just started taking to Introduction to Analysis and I find it very much difficult to prove such a thing. Could anyone please help me?

Answer 1

You can simply take the logaritms of both members: $$\log_a(a^r)<\log_a(a^s)$$ and because $a>1$ you have that it is true if and only if $r<s$.