Question

Solve
${\log }_{5}x+{\log }_{x}5=5/2$

Answer 1

We have,

${\log }_{5}x+{\log }_{x}5=\frac{5}{2}$

$\Rightarrow {\log }_{5}x+\frac{1}{{\log }_{5}x}=\frac{5}{2}\therefore {\log }_{a}x=\frac{1}{{\log }_{x}a}$

let ${\log }_{5}x=y$

then,

$y+\frac{1}{y}=\frac{5}{2}$

$\Rightarrow y^2+1=\frac{5}{2}y$

$\Rightarrow 2y^2+2=5y$

$\Rightarrow 2y^2+2-5y=0$

$\Rightarrow 2y^2-5y+2=0$

$\Rightarrow 2y^2-4y-y+2=0$

$\Rightarrow 2y(y-2)-1(y-2)=0$

$\Rightarrow (y-2)(2y-1)=0$

$\Rightarrow y-2=0,2y-1=0$

$\Rightarrow y=2,y=\frac{1}{2}$

Now taking

${\log }_{5}x=y$

${\log }_{5}x=2$

$x=5^2$

$x=25$

${\log }_{5}x=y$

${\log }_{5}xx=\frac{1}{2}$

$x=5^{\frac{1}{2}}$

$x=\sqrt{5}$

Hence, this is the answer.