Product of all the solution of the equation&n | vedclass

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Question

$x^{1+\log _{10} x}=100000 x$

$\Rightarrow x \cdot x^{\log _{10} x}=100000 x$

$\Rightarrow x\left(x^{\log _{10} x}-10^{5}ight)=0$

$x 
eq 0

Vedclass · Accepted Answer

$1$

Vedclass · Answer

$x^{1+\log _{10} x}=100000 x$

$\Rightarrow x \cdot x^{\log _{10} x}=100000 x$

$\Rightarrow x\left(x^{\log _{10} x}-10^{5}ight)=0$

$x 
eq 0, \quad x^{\log _{10} x}=10^{5}$

$\Rightarrow \log _{10} x=5 \log _{x}(10)$

$\Rightarrow(\log x)^{2}=5$

$\Rightarrow \log x=\pm \sqrt{5}$

$ 	herefore x=10^{\sqrt {5}} $  and $10^{-\sqrt{5}}$

Product of solution $=10^{\sqrt{5}} \cdot 10^{-\sqrt{5}}=1$

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

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