Logarithm/Base/Rules/Fact/Proof/Exercise

Prove that for the logarithm to base ${}b$ the following calculation rules hold.

1. We have ${}\log _{b}(b^{x})=x$ and ${}b^{\log _{b}(y)}=y$, ie, the logarithm to base ${}b$ is the inverse to the exponential function to the 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

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

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