Question

suppose, log3 x = log12 y = a, where x, y are positive numbers. if g is the geometric mean of x and y, and log6 g is equal to

Answer 1

Given, $log_3x=log_{12}y=a$
so, $log_3x=a$
we know, $log_NM=a\implies M=N^a$
$x=3^a$.......(1)

similarly, $log_{12}y=a$
$y=12^a=(3\times)^a=3^a.4^a$....(2)

A/C to question,
g is geometric mean of x and y,
e.g., $g=\sqrt{xy}$
from equations (1) and (2),
$g=\sqrt{3^a.3^a.4^a}$
$g=\sqrt{(3^a)^2.(2^a)^2}$
$g=3^a.2^a=6^a$
now taking log base 6 both sides,
$log_6g=log_66^a=a$

hence, $log_6g=a$