Question Number 76009 by Rio Michael last updated on 22/Dec/19
$${hiw}\:{do}\:{i}\:{solve} \\ $$$$\mathrm{2}^{{x}} \:=\:\mathrm{4}{x}? \\ $$
Answered by mr W last updated on 22/Dec/19
$$\mathrm{2}^{{x}} =\mathrm{4}{x} \\ $$$${e}^{{x}\mathrm{ln}\:\mathrm{2}} =\mathrm{4}{x} \\ $$$${xe}^{−{x}\mathrm{ln}\:\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left(−{x}\mathrm{ln}\:\mathrm{2}\right){e}^{−{x}\mathrm{ln}\:\mathrm{2}} =−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}} \\ $$$$−{x}\mathrm{ln}\:\mathrm{2}={W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}\right) \\ $$$$\Rightarrow{x}=−\frac{{W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}\right)}{\mathrm{ln}\:\mathrm{2}}=\begin{cases}{−\frac{−\mathrm{0}.\mathrm{21481112}}{\mathrm{ln}\:\mathrm{2}}=\mathrm{0}.\mathrm{3099}}\\{−\frac{−\mathrm{2}.\mathrm{77258872}}{\mathrm{ln}\:\mathrm{2}}=\mathrm{4}}\end{cases} \\ $$
Commented by vishalbhardwaj last updated on 22/Dec/19
$$\mathrm{W}\:=\:??????? \\ $$
Commented by mr W last updated on 22/Dec/19
$${Lambert}\:{W}\:{function}: \\ $$$${W}\left({x}\right)×{e}^{{W}\left({x}\right)} ={x} \\ $$
Commented by Rio Michael last updated on 24/Dec/19
$${sir}\:{i}\:{wanna}\:{know}\:{abt}\:{this}\:{why} \\ $$$${are}\:{you}\:{taking}\:{e}\:{only}\:{one}\:{one}\:{side}? \\ $$
Commented by Rio Michael last updated on 24/Dec/19
$${okay}\:{sir}\:{and}\:{in}\:{that}\:{form}\:{we}\:{manupulate}\: \\ $$$${and}\:{solve}\:{for}\:{x}. \\ $$
Commented by mr W last updated on 24/Dec/19
$${to}\:{use}\:{lambert}\:{W}\:{function}\:{we} \\ $$$${should}\:{arrange}\:{the}\:{equation}\:{into} \\ $$$${the}\:{form}\:\left(…\right){e}^{\left(…\right)} =\left(……\right) \\ $$