Product of all solution of equation {x^{1 + {{log }_{10}}x}} = 100000x is

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1.Relation and Function

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

A

$10$

B

$10^5$

C

$10^{-5}$

D

$1$

Solution

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

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

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

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

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

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

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

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

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

Standard 12

Mathematics

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