If $$\log 2 = 0.3010\,$$   and $$\log 3 = 0.4771,\,$$   the value of $${\log _5}512$$   = ?

$$\eqalign{ & {\log _5}512 \cr & = \frac{{\log 512}}{{\log 5}} \cr & = \frac{{\log {2^9}}}{{\log \left( {\frac{{10}}{2}} \right)}} \cr & = \frac{{9\log 2}}{{\log 10 - \log 2}} \cr & = \frac{{\left( {9 \times 0.3010} \right)}}{{1 - 0.3010}} \cr & = \frac{{2.709}}{{0.699}} \cr & = \frac{{2709}}{{699}} \cr & = 3.876 \cr} $$