Question

Solve for $x$:

${\log }_{x}2.{\log }_{\frac{x}{16}}2={\log }_{\frac{x}{64}}2$.

Answer 1

Given,

${\log }_{x}2.{\log }_{\frac{x}{16}}2={\log }_{\frac{x}{64}}2$

or, ${\log }_{2}x.{\log }_2\frac{x}{16}={\log }_2\frac{x}{64}$ [ Taking reciprocal each sides]

or, ${\log }_{2}x.({\log }_{2}x-{\log }_{2}16)={\log }_{2}x-{\log }_{2}64$

or, ${\log }_{2}x.({\log }_{2}x-4)={\log }_{2}x-6$

or, $({\log }_{2}x{)}^2-5{\log }_{2}x+6=0$

or, $({\log }_{2}x-3)({\log }_{2}x-2)=0$

or, ${\log }_{2}x=3,{\log }_{2}x=2$

or, $x=2^3,2^2$

or, $x=8,4$.