Logarithm/Base/Rules/Fact

The logarithms to base ${}b$ fulfill the following rules.

1. We have ${}\log _{b}(b^{x})=x$ and ${}b^{\log _{b}(y)}=y$, this means that the logarithm to Base ${}b$ is the inverse function for the exponential function to base ${}b$.
2. We have ${}\log _{b}(y\cdot z)=\log _{b}y+\log _{b}z$.
3. We have ${}\log _{b}y^{u}=u\cdot \log _{b}y$ for ${}u\in \mathbb {R}$.
4. We have

   ${}\log _{a}y=\log _{a}{\left(b^{\log _{b}y}\right)}=\log _{b}y\cdot \log _{a}b\,.$

Write a proof