Question Number 34792 by mondodotto@gmail.com last updated on 11/May/18
$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{{x}} \\ $$$$\mathrm{4}\boldsymbol{{x}}=\mathrm{2}^{\boldsymbol{{x}}} \\ $$
Answered by Joel579 last updated on 11/May/18
Commented by Joel579 last updated on 11/May/18
$$\mathrm{Real}\:\mathrm{solution} \\ $$$${x}\:=\:\mathrm{0}.\mathrm{3099} \\ $$$${x}\:=\:\mathrm{4} \\ $$
Answered by mr W last updated on 13/Jan/19
$$\mathrm{4}{x}={e}^{{x}\mathrm{ln}\:\mathrm{2}} \\ $$$${xe}^{−{x}\mathrm{ln}\:\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left(−{x}\mathrm{ln}\:\mathrm{2}\right){e}^{−{x}\mathrm{ln}\:\mathrm{2}} =−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}} \\ $$$$\Rightarrow−{x}\:\mathrm{ln}\:\mathrm{2}={W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}\right) \\ $$$$\Rightarrow{x}=−\frac{{W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}\right)}{\mathrm{ln}\:\mathrm{2}}=\begin{cases}{−\frac{−\mathrm{0}.\mathrm{2148}}{\mathrm{ln}\:\mathrm{2}}=\mathrm{0}.\mathrm{3099}}\\{−\frac{−\mathrm{2}.\mathrm{7726}}{\mathrm{ln}\:\mathrm{2}}=\mathrm{4}}\end{cases} \\ $$