\log _4\left(0.2\right)\times \log _{125}\left(0.125\right)

asked by guest
on Nov 22, 2024 at 10:23 am



You asked:

Evaluate the expression: \(\log_{4}\left( 0.2 \cdot \log_{125}\left( 0.125 \right) \right)\)

MathBot Answer:

\[\log_{4}\left( 0.2 \cdot \log_{125}\left( 0.125 \right) \right) = \frac{\log{\left(\frac{\log{\left(8 \right)}}{5 \log{\left(125 \right)}} \right)} + i \pi}{\log{\left(4 \right)}}\]

\[\begin{aligned}\log_{4}\left( 0.2 \cdot \log_{125}\left( 0.125 \right) \right)& = \log{\left(\frac{2}{10} \cdot \log{\left(\frac{125}{1000} \right)} \right)}\\& = \log{\left(\frac{2}{10} \cdot \log{\left(\frac{1}{8} \right)} \right)}\\& = \log{\left(\frac{2}{10} \cdot - \cdot \frac{1}{\log{\left(125 \right)}} \cdot \log{\left(8 \right)} \right)}\\& = \log{\left(\frac{1}{5} \cdot - \cdot \frac{1}{\log{\left(125 \right)}} \cdot \log{\left(8 \right)} \right)}\\& = \log{\left(- \frac{1}{5} \cdot \frac{1}{\log{\left(125 \right)}} \cdot \log{\left(8 \right)} \right)}\\& = \frac{1}{\log{\left(4 \right)}} \cdot \left(\pi \cdot i + \log{\left(\frac{1}{5} \cdot \frac{1}{\log{\left(125 \right)}} \cdot \log{\left(8 \right)} \right)}\right)\end{aligned}\]

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