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Prove two formulas for change base of logarithm

Prove:

  1. \displaystyle{\log_b x = \log_b a \log_a x}.
  2. \displaystyle{\log_b x = \frac{\log_a x}{\log_a b}}.

  1. 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*}

  2. 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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