Question

The product of all the solutions of the equation $x^{1+lo{g}_{10}x}=100000x$ is -

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

Answer 1

The correct option is D 1

Given,

$x^{1+{\log }_{10}\left(x\right)}=100000x$

${\log }_{10}\left(x^{1+{\log }_{10}\left(x\right)}\right)={\log }_{10}\left(100000x\right)$

$(1+{\log }_{10}\left(x\right)){\log }_{10}\left(x\right)={\log }_{10}\left(100000x\right)$

$(1+{\log }_{10}\left(x\right)){\log }_{10}\left(x\right)={\log }_{10}\left(x\right)+5$

$\Rightarrow (1+u)u=u+5$ upon substitution

$\Rightarrow u^2=5$

$\therefore u=\pm \sqrt{5}$

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

$\therefore x={10}^{\sqrt{5}},\frac{1}{{10}^{\sqrt{5}}}$

Therefore, the product of roots, $={10}^{\sqrt{5}}\times \frac{1}{{10}^{\sqrt{5}}}=1$