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 $u\in ℝ$.
4. We have

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

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