Question

The product of all the solution of the equation $x^{1+{\log }_{10}x}=100000x$ is

• A. 10
• B. ${10}^5$
• C. ${10}^{-5}$
• D. 1

Answer 1

The correct option is D 1

$x^{1+{\log }_{10}x}=100000x$

$\Rightarrow x\cdot x^{{\log }_{10}x}=100000x$

$\Rightarrow x(x^{{\log }_{10}x}-{10}^5)=0$

$x\ne 0,x^{{\log }_{10}x}={10}^5$

$\Rightarrow {\log }_{10}x=5{\log }_x\left(10\right)$

$\Rightarrow {\left(\log x\right)}^2=5$

$\Rightarrow \log x=\pm \sqrt{5}$

$\therefore x={10}^{\sqrt{5}}$ and ${10}^{-\sqrt{5}}$

Product of solution $={10}^{\sqrt{5}}\cdot {10}^{-\sqrt{5}}=1$