If _102 = a and _103 = b, then _512 = ?

Answer: Option D

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