Answer to Question #223279 in Algebra for Gartiffer

Question

GIven that log₃6=m, and log₆5=n express log₃10 in terms of m and n

Answer 1

GIven, $log_36$=m, and $log_65$ =n.

$log_36=m\\ \frac{log6}{log3}=m\space--------(1)\\ \frac{log(2×6)}{log3}=m\\ \frac{log2+log3}{log3}=m\\ \frac{log2}{log3}+\frac{log3}{log3}=m\\ \frac{log2}{log3}=m-1\\ And,\\ log_65=n\\ \frac{log5}{log6}=n\space---------(2)\\ \text{By multiple (1) and (2), we get}\\ \frac{log6}{log3}\frac{log5}{log6}=\frac{log5}{log3}=mn\space ----(3)\\ Then,\\ log_310=\frac{log10}{log3}\\ =\frac{log(2×5)}{log3}\\ =\frac{log2+log5}{log3}\\ =\frac{log2}{log3}+\frac{log5}{log3}\\ =m-1+mn\space \space[from \space(3)]\\ =m(1+n)-n$