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Question Number 172020 by Mikenice last updated on 23/Jun/22

solve:  2^x =10x

$${solve}: \\ $$$$\mathrm{2}^{{x}} =\mathrm{10}{x} \\ $$

Answered by mr W last updated on 23/Jun/22

e^(xln 2) =10x  (−xln 2)e^(−xln 2) =−((ln 2)/(10))  −xln 2=W(−((ln 2)/(10)))  ⇒x=−(1/(ln 2))W(−((ln 2)/(10)))= { ((0.107755)),((5.877011)) :}

$${e}^{{x}\mathrm{ln}\:\mathrm{2}} =\mathrm{10}{x} \\ $$$$\left(−{x}\mathrm{ln}\:\mathrm{2}\right){e}^{−{x}\mathrm{ln}\:\mathrm{2}} =−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{10}} \\ $$$$−{x}\mathrm{ln}\:\mathrm{2}={W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{10}}\right) \\ $$$$\Rightarrow{x}=−\frac{\mathrm{1}}{\mathrm{ln}\:\mathrm{2}}{W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{10}}\right)=\begin{cases}{\mathrm{0}.\mathrm{107755}}\\{\mathrm{5}.\mathrm{877011}}\end{cases} \\ $$

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