If-log-12-27-a-express-log-6-16-in-terms-of-a-

Question Number 113811 by Aina Samuel Temidayo last updated on 15/Sep/20

If \log_{12} 27 = a, express \log_{6} 16 in terms

of a .

Answered by bemath last updated on 15/Sep/20

\frac{\ln 27}{\ln 12} = a \Rightarrow \frac{3 \ln 3}{a} = \ln 12

\frac{3 \ln 3}{a} = \ln 3 + 2 \ln 2 \to 2 \ln 2 = \frac{3 \ln 3}{a} - \ln 3

\Leftrightarrow \frac{\ln 16}{\ln 6} = \frac{4 \ln 2}{\ln 2 + \ln 3} = \frac{\frac{2 (3 - a) \ln 3}{a}}{\frac{(3 - a) \ln 3}{2 a} + \ln 3}

= \frac{\frac{6 - 2 a}{a}}{\frac{3 - a + 2 a}{2 a}} = \frac{6 - 2 a}{a} \times \frac{2 a}{3 + a} = \frac{12 - 4 a}{3 + a}

Commented by Aina Samuel Temidayo last updated on 15/Sep/20

= \frac{4 (3 - a)}{3 + a}

Thanks .