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1. If $\log 2 = 0.3010$ and $\log 3 = 0.4771,$ the value of $\log_{5} 512$ = ?

A. 2.870

B. 2.967

C. 3.876

D. 3.912

Solution:

\begin{aligned} \log_{5} 512 \\ = \frac{\log 512}{\log 5} \\ = \frac{\log 2^{9}}{\log (\frac{10}{2})} \\ = \frac{9 \log 2}{\log 10 - \log 2} \\ = \frac{(9 \times 0.3010)}{1 - 0.3010} \\ = \frac{2.709}{0.699} \\ = \frac{2709}{699} \\ = 3.876 \end{aligned}

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