Question

If the third term in the expansion ${(x+x^{{\log }_{5}x})}^5$ is $2$, then $x$ equals

• A. $1/5,5$
• B. $1/5,1/\sqrt{5}$
• C. $\sqrt{5},5$
• D. $1/\sqrt{5},5$

Answer 1

Answer: (B)

$1/5,1/\sqrt{5}$

$T_{2+1}$
$={}^5{C}_2{x}^{3+2lo{g}_5\left(x\right)}$
$=10x^{3+2lo{g}_5\left(x\right)}$
$=2$
$x^{3+2lo{g}_5\left(x\right)}=\frac{1}{5}$
$lo{g}_5\left(x\right)(3+2lo{g}_5(x\left)\right)=-1$
$2log(x{)}^2+3log(x)+1=0$
$2log(x{)}^2+2log(x)+log(x)+1=0$
$\left(2log\right(x)+1)\left(log\right(x)+1)=0$
$log\left(x\right)=-1$
$x=\frac{1}{5}$ and $log\left(x\right)=\frac{-1}{2}$
$x=\frac{1}{\sqrt{5}}$