log12(4) + ( ln(3) / ln(12) )

asked by guest
on Sep 21, 2024 at 6:39 am



You asked:

Evaluate the expression: \(\log_{12}\left( 4 \right) + \frac{\ln\left( 3 \right)}{\ln\left( 12 \right)}\)

MathBot Answer:

\[\log_{12}\left( 4 \right) + \frac{\ln\left( 3 \right)}{\ln\left( 12 \right)} = \frac{\log{\left(3 \right)}}{\log{\left(12 \right)}} + \frac{\log{\left(4 \right)}}{\log{\left(12 \right)}} \approx 1.0\]

\[\begin{aligned}\log_{12}\left( 4 \right) + \frac{\ln\left( 3 \right)}{\ln\left( 12 \right)}& = \log{\left(4 \right)} + \log{\left(3 \right)} \cdot \frac{1}{\log{\left(12 \right)}}\\& = \frac{1}{\log{\left(12 \right)}} \cdot \log{\left(4 \right)} + \log{\left(3 \right)} \cdot \frac{1}{\log{\left(12 \right)}}\\& = \frac{1}{\log{\left(12 \right)}} \cdot \log{\left(4 \right)} + \frac{1}{\log{\left(12 \right)}} \cdot \log{\left(3 \right)}\\& = \frac{1}{\log{\left(12 \right)}} \cdot \log{\left(3 \right)} + \frac{1}{\log{\left(12 \right)}} \cdot \log{\left(4 \right)}\end{aligned}\]

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