Question

prove that there does that not exits solution of the equation $\frac{\log 5+\log (x^2+1)}{2\log (x-2)}=1$

Answer 1

By definition of log, we must have
$log\left\{5\right(x^2+1\left)\right\}=log(x-2{)}^2$
or $5x^2+5=x^2-4x+4$
or $4x^2$ $+4x+1=0$
or $(2x+1{)}^2$ = 0
$\therefore x=\frac{-1}{2}$
But we must have $x>2$, hence no sloution.