Prove two formulas for change base of logarithm

Prove:

Proof. The proof is a direct computation,
$\begin{align*} \log_b x &= \frac{\log x}{\log b} \\ &= \left( \frac{\log x}{\log b} \right) \left( \frac{\log a}{\log a} \right) \\ &= \left( \frac{\log a}{\log b} \right) \left( \frac{\log x}{\log a} \right) \\ &= \left( \log_a x \right) \left( \log_b a \right). \qquad \blacksquare \end{align*}$
Proof. Again, this is a direct computation,
$\begin{align*} \log_b x &= \frac{\log x}{\log b} \\ &= \left( \frac{\log x}{\log b} \right) \left( \frac{\log a}{\log a} \right) \\ &= \left( \frac{\log x}{\log a} \right) \left( \frac{\log b}{\log a} \right)^{-1} \\ &= \log_a x (\log_a b)^{-1} \\ &= \frac{\log_a x}{\log_a b}. \qquad \blacksquare \end{align*}$

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