Question

Given: log2 = a, log7 = b. Find: log56

Answer 1

$\bf \begin{array}{llll} \textit{logarithm of factors} \\\\ log_a(xy)\implies log_a(x)+log_a(y) \end{array} ~\hfill \begin{array}{llll} \textit{Logarithm of exponentials} \\\\ log_a\left( x^b \right)\implies b\cdot log_a(x) \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf log(56)~~ \begin{cases} 56=2\cdot 2\cdot 2\cdot 7\\ \qquad 2^2\cdot 2\cdot 7 \end{cases}\implies log(2^2\cdot 2\cdot 7) \\\\\\ log(2^2)+\stackrel{a}{log(2)}+\stackrel{b}{log(7)}\implies 2\stackrel{a}{log(2)}+a+b\implies 2a+a+b\implies 3a+b$