If 2 = 0.3010\, and 3 = 0.4771,\, the value of _5512 = ?

Answer: Option C

$$\begin{array}{rl}& {\log }_{5}512\\ & =\frac{\log 512}{\log 5}\\ & =\frac{\log 2^9}{\log \left(\frac{10}{2}\right)}\\ & =\frac{9\log 2}{\log 10-\log 2}\\ & =\frac{\left(9\times 0.3010\right)}{1-0.3010}\\ & =\frac{2.709}{0.699}\\ & =\frac{2709}{699}\\ & =3.876\end{array}$$