x-3-3-x-how-do-i-pls-use-lambert-to-find-x-3-

Question Number 151403 by 7770 last updated on 20/Aug/21

$$\:{x}^{\mathrm{3}} =\mathrm{3}^{{x}} \\ $$$${how}\:{do}\:{i}\:{pls}\:{use}\:{lambert}\:{to}\: \\ $$$$\:{find}\:{x}=\mathrm{3} \\ $$

Commented by mr W last updated on 20/Aug/21

x=3^(x/3) x=e^((x/3)ln 3) xe^(−(x/3)ln 3) =1 (−(x/3)ln 3)e^(−(x/3)ln 3) =−((ln 3)/3) −(x/3)ln 3=W(−((ln 3)/3)) ⇒x=−(3/(ln 3))W(−((ln 3)/3)) = { ((−(3/(ln 3))×(−1.098612289)=3)),((−(3/(ln 3))×(−0.907473042)=2.478053)) :}

$${x}=\mathrm{3}^{\frac{{x}}{\mathrm{3}}} \\ $$$${x}={e}^{\frac{{x}}{\mathrm{3}}\mathrm{ln}\:\mathrm{3}} \\ $$$${xe}^{−\frac{{x}}{\mathrm{3}}\mathrm{ln}\:\mathrm{3}} =\mathrm{1} \\ $$$$\left(−\frac{{x}}{\mathrm{3}}\mathrm{ln}\:\mathrm{3}\right){e}^{−\frac{{x}}{\mathrm{3}}\mathrm{ln}\:\mathrm{3}} =−\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{3}} \\ $$$$−\frac{{x}}{\mathrm{3}}\mathrm{ln}\:\mathrm{3}={W}\left(−\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{3}}\right) \\ $$$$\Rightarrow{x}=−\frac{\mathrm{3}}{\mathrm{ln}\:\mathrm{3}}{W}\left(−\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{3}}\right) \\ $$$$=\begin{cases}{−\frac{\mathrm{3}}{\mathrm{ln}\:\mathrm{3}}×\left(−\mathrm{1}.\mathrm{098612289}\right)=\mathrm{3}}\\{−\frac{\mathrm{3}}{\mathrm{ln}\:\mathrm{3}}×\left(−\mathrm{0}.\mathrm{907473042}\right)=\mathrm{2}.\mathrm{478053}}\end{cases} \\ $$

Commented by 7770 last updated on 20/Aug/21