Question

Solve the following equation:
$\frac{\log \sqrt{x+7}-\log 2}{\log 8-\log (x-5)}=-1$

Answer 1

$\frac{log\sqrt{x+7}-log2}{log8-log(x-5)}=-1log\sqrt{x+7}-log2=-log8+log(x-5)\frac{1}{2}log(x+7)=-2log2+log(x-5)log(x+7)=2(-log4+log(x-5\left)\right)(x+7)={\left(\frac{x-5}{4}\right)}^{2}16x+112=x^2-10x+25x^2-26x-87=0x=29,-3$

But at $x=-3$, $x-5<0$ which is not valid.

$\therefore x=29$ is only solution.