Evaluate $\displaystyle D_z[3^z\ln(z+5)]$

I used the product rule in combination with the chain rule.

Our professor taught us that in order to solve $\displaystyle D_z3^z$ we must rewrite $\displaystyle 3^z$ as $\displaystyle e^{ln3^z}$. So, using the chain rule, i found that $\displaystyle D_z\ln(z+5) \ = \ \ln3 \ \times e^{z\ln3}$.

Continuing on with the product rule...

$\displaystyle 3^z \ \times \ D_z[\ln(z+5)] \ + \ \ln(z+5) \ \times \ D_z[3^z]$.

Which, rewritten, looks like:

$\displaystyle 3^z \ \times \ \frac{1}{z+5} \ + \ \ln(z+5) \ \times \ \ln3 \ \times \ e^{zln3}$.

Now I am not sure if there is a way to rewrite what I have above to make it match this, but the answer we were given is $\displaystyle 3^z[\frac{1}{z+5} \ + \ \ln(z+5)\ln3]$.

Am I doing something wrong in calculating my derivative, or is there a way to rewrite my answer that I'm not seeing?