Under $a > 0$, $b > 0$, $x \neq 0$, if $a^x = b^x$ then—

Since $x\neq 0$, raise both sides to the power $1/x$: $\left(a^x\right)^{1/x}=\left(b^x\right)^{1/x}$ gives $a=b$. Equal powers with the same nonzero exponent require equal positive bases.