Need help with expressing this logarithm

Question

Express $log_3(a^2 + \sqrt{b})$ in terms of m and k where $m = log_{3}a$

$k = log_{3}b$

Given this information I made $a = 3^m$

$b = 3^k$

Therefore = $log_{3} ((3^m)^2 + (3^k))^{\frac{1}{2}}$

= $log_{3} (3^{2m} + 3^{\frac{k}{2}})$

I don't know if I'm done or there is still more things I can simplify. Can anyone help please, thanks

Answer 1

Since there is no expansion for $\log(a+b)$, hence, you could stop there. If you need to get an approximation(sometimes in computer science), you could use $\log(a+b)=\log a+\log(1+b/a)$, then you have following: $$2m+\log_3(1+3^{k/2-2m}).$$

Answer 2

Alternatively, $\log_3(9^m+\sqrt3^k)$, but this makes little difference.

You can't decompose a sum inside a logarithm.