Question

Solving for x of the given equation ${\log }_5(5^{\frac{1}{x}}+125)={\log }_{5}6+1+\frac{1}{2x}$ we get two solutions.Let they be m, n. Find $\frac{1}{m\ast n}$ ?

Answer 1

${\log }_5(5^{\frac{1}{x}}+125)={\log }_{5}6+1+\frac{1}{2x}$
${\log }_5\left(\frac{5^{\frac{1}{x}}+125}{6}\right)=1+\frac{1}{2x}$
$5^{1+\frac{1}{2x}}=\frac{5^{\frac{1}{x}}+125}{6}$
$\Rightarrow 5^{1/x}-30\left(5^{1/2x}\right)+125=0$
Substitute $5^{1/2x}=t$
$\Rightarrow t^2-30t+125=0$
$\Rightarrow (t-25)(t-5)=0$
$\Rightarrow t=25,5$
$\Rightarrow 5^{1/2x}=25,5^{1/2x}=5$
$\Rightarrow \frac{1}{2x}=2,\frac{1}{2x}=1$
$\Rightarrow x=\frac{1}{2},\frac{1}{4}$
From the given eqn , it follows that
$5^{1/x}+125>0$
$5^{1/x}>-5^3$
Clearly, $x=\frac{1}{2},\frac{1}{4}$ satisfies above inequality.
Thus, $x=\frac{1}{2},\frac{1}{4}$ are the solution of given eqn

Hence, $\frac{1}{\frac{1}{2}\times \frac{1}{4}}=\frac{1}{\frac{1}{8}}=8$