So, here is the clear formulation of the problem: let $(x_n) $ be a convergent sequence of positive numbers, with $x_n \to c$. I want to prove that the sequence $(y_n) $, with $y_n=a^{x_n} $, tends to $a^c$
I will use the proof by contradiction. Suppose that $y_n \to, l$ with $l \neq a^c$.
But, we know that if $y_n \to l$, then $\ln{y_n} \to \ln l$. That means that:
$$\lim{\ln{y_n}} =\ln l$$ $$\lim{\ln{a^{x_n}} } =\ln l$$ $$\lim{(x_n\ln{a}) } =\ln l$$ $$\ln{a}(\lim{x_n}) =\ln l$$ $$c\ln{a} =\ln l$$ $$\ln{a^c} =\ln l$$
And now, from the injectivity of logarithm function it follows that $l=a^c$, which is a contradiction. Hence, Q. E. D.
Please, can you tell me if my proof is a correct one?
This is correct provided you're allowed to assume that $\lim (\ln y_n) = \ln (\lim y_n)$.
Your proof structure could be simplified, though: you gave a self-contained proof that $l=a^c$, but surrounded it unnecessarily with 'suppose $l \ne a^c$' and 'contradiction'. If you just didn't assume the result was false (and therefore never reached a contradiction) then you'd still have a proof.