Question

The value of $x$ in the expression ${\left(x+x^{{\log }_{10}x}\right)}^5$ if the third term in the expansion is 1000000 is

• A. $10,10^{-3/2}$
• B. 100 or $10^{-3/2}$
• C. 10 or $10^{-5/2}$
• D. None of these

Answer 1

Answer: (C)

$\log x$ is defined only when $x>0$
Now, the 3rd term in the expansion is $1000000$
$\Rightarrow {}^5{C}_2{x}^{5-2}\cdot {\left(x^{{\log }_{10}x}\right)}^2=1000000$
$\Rightarrow x^{3+2{\log }_{10}x}=10^5$
Taking logarithm of both sides, we get
$\left(3+2{\log }_{10}x\right){\log }_{10}x=5$
$\Rightarrow 2y^2+3y-5=0,$ where ${\log }_{10}x=y$
$\Rightarrow (y-1)(2y+5)=0$
$\Rightarrow y=1$ or $-5/2$
$\Rightarrow {\log }_{10}x=1$ or $-5/2$
$\Rightarrow x=10^1=10$ or $10^{-5/2}$