Can you substitute functions in Big-$\Theta$ notation?
Bit $\Theta$ gives big possibilities. Let's write down what we have, assuming appropriate conditions on constants and infinity neighbourhood:
$$f(n)=\Theta(\log n) \Leftrightarrow a_1 \log n \leqslant f \leqslant a_2 \log n$$
$$g(n)=\Theta(n+\log n) \Leftrightarrow b_1 (n+\log n) \leqslant g \leqslant b_2 (n+\log n)$$
Now we combine from above inequalities
$$ c_1 (n+f) \leqslant b_1n+\frac{b_1}{a_2}f\leqslant g \leqslant b_2 n + \frac{b_2}{a_1}f \leqslant c_2 (n+f) $$
which gives $g(n) = \Theta(n + f(n))$.