How to prove "if $2^x=2^y$ so $x=y$"

Question

I got this to prove and I don't know how. I tried everything but I just don't know how to prove it. thanks for helping.

Answer 1

The function $f:\mathbb{R}\rightarrow \mathbb{R}$, $f(x)=2^x$ is injective (for instance because it is monotone increasing). Since $2^x=2^y$ means $f(x)=f(y)$, this implies $x=y$.

Answer 2

Taking the logarithm on both sides, we get$$x\ln(2)=y\ln(2)$$