Question

Solve for $x:\frac{{\log }_{10}(x-3)}{{\log }_{10}(x^2-21)}=\frac{1}{2}$

Answer 1

$\frac{{\log }_{10}(x-3)}{{\log }_{10}(x^2-21)}=\frac{1}{2}$

$\Rightarrow {\log }_{(x^2-21)}(x-3)=\frac{1}{2}$

$\Rightarrow x-3=\sqrt{x^2-21}$

By squaring both sides, we get

$\Rightarrow {(x-3)}^2=x^2-21$

$\Rightarrow x^2-6x+9=x^2-21$

$\Rightarrow -6x+9=-21$

$\Rightarrow -6x=-21-9=-30$

$\Rightarrow x=\frac{-30}{-6}=5$

$\therefore x=5$