$$\eqalign{ & {\text{If}}\,\,{\log _{10}}2 = a\,{\text{and}} \cr & {\log _{10}}3 = b,\,{\text{then}} \cr & {\log _5}12 = ? \cr}$$

Explanation:-

$$\eqalign{ & {\log _5}12 = {\log _5}\left( {3 \times 5} \right) \cr & = {\log _5}3 + {\log _5}4 \cr & = {\log _5}3 + 2{\log _5}2 \cr & = \frac{{{{\log }_{10}}2}}{{{{\log }_{10}}5}} + \frac{{2{{\log }_{10}}2}}{{{{\log }_{10}}5}} \cr & = \frac{{{{\log }_{10}}3}}{{{{\log }_{10}}10 - {{\log }_{10}}2}} + \frac{{2{{\log }_{10}}2}}{{{{\log }_{10}}10 - {{\log }_{10}}2}} \cr & = \frac{b}{{1 - a}} + \frac{{2a}}{{1 - a}} \cr & = \frac{{2a + b}}{{1 - a}} \cr}$$