(1)

\begin{align*} \log M+\log N & =\log\left(\exp(\log M+\log N)\right)\\ & =\log\left(\exp(\log M)\exp(\log N)\right)\\ & =\log\left(MN\right) \end{align*}

(2)

\begin{align*} \log M^{r} & =\log\exp^{r}(\log M)\\ & =\log\exp(r\log M)\\ & =r\log M \end{align*}

(3)

\begin{align*} \log_{a}b & =\log_{a}c^{\log_{c}b}\\ & =\log_{a}c^{\log_{c}a\frac{\log_{c}b}{\log_{c}a}}\\ & =\log_{a}a^{\frac{\log_{c}b}{\log_{c}a}}\\ & =\frac{\log_{c}b}{\log_{c}a} \end{align*}

(4)

\begin{align*} \log_{a}b & =\frac{\log_{b}b}{\log_{b}a}\\ & =\frac{1}{\log_{b}a} \end{align*}