How can I find $\Theta(log(m_1)+...+log(m_k))$ as related to $m$?
The $\log$ function is concave so, by Jensen's inequality,
$$\frac{1}{k}\sum_{i=1}^k\log m_i \leq \log\Big(\frac1k\sum_{i=1}^k m_i\Big)$$
i.e.,
$$\sum_{i=1}^k \log m_i \leq k\log\frac{m}{k} = k\,(\log m - \log k)\,,$$
which is basically what you got by bounding $\log m_i\leq \log m$.  We can't do any better than this because we could have $m_1 = \dots = k_m = \tfrac{m}{k}$, in which case $\sum_i\log m_i = k\log\tfrac{m}{k}$.

However, if $k$ is a fixed constant, then $k\log\tfrac{m}{k}=O(\log m)$.