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Question Number 17742 by tawa tawa last updated on 10/Jul/17

$${\mathrm{x}}^3=3^{\mathrm{x}},\mathrm{find}\mathrm{x}.$$

Commented by 1kanika# last updated on 10/Jul/17

$$\mathrm{x}=3$$

Answered by mrW1 last updated on 10/Jul/17

$$\mathrm{x}=3^{\frac{\mathrm{x}}{3}}={\mathrm{e}}^{\frac{\mathrm{x}}{3}\ln 3}$$$${\mathrm{xe}}^{-\frac{\mathrm{x}}{3}\ln 3}=1$$$$(-\frac{\mathrm{x}}{3}\ln 3){\mathrm{e}}^{-\frac{\mathrm{x}}{3}\ln 3}=-\frac{\ln 3}{3}$$$$-\frac{\mathrm{x}}{3}\ln 3=\mathrm{W}(-\frac{\ln 3}{3})$$$$\Rightarrow \mathrm{x}=-\frac{3}{\ln 3}\mathrm{W}(-\frac{\ln 3}{3})=\{\begin{array}{l}2.4...\\ 3\end{array}$$

Commented by tawa tawa last updated on 10/Jul/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{i}\mathrm{understand}\mathrm{it}.\mathrm{i}\mathrm{really}\mathrm{appreciate}.$$

Commented by mrW1 last updated on 10/Jul/17

$$\mathrm{solve}{\mathrm{x}}^2=3^{\mathrm{x}}$$

Commented by tawa tawa last updated on 10/Jul/17

$${\mathrm{x}}^2=3^{\mathrm{x}}$$$$3^{\frac{\mathrm{x}}{2}}=\mathrm{x}$$$${\mathrm{e}}^{\frac{\mathrm{x}}{2}\ln 3}=\mathrm{x}$$$$1={\mathrm{xe}}^{-\frac{\mathrm{xln}3}{2}}$$$$-\frac{\ln 3}{2}=(-\frac{\mathrm{xln}3}{2}){\mathrm{e}}^{(-\frac{\mathrm{xln}3}{2})}$$$$-\frac{\mathrm{xln}3}{2}={\mathrm{We}}^{(-\frac{\ln 3}{2})}$$$$\mathrm{x}=\frac{-{2\mathrm{We}}^{(-\frac{\ln 3}{2})}}{\ln 3}$$$$\mathrm{x}=\frac{-{2\mathrm{We}}^{\left(\frac{\ln 3}{2}\right)}}{\ln 3}$$$$\mathrm{x}=\frac{-2\times 0.376839}{1.09861}$$$$\mathrm{x}=\frac{-0.753674}{1.09861}$$$$\mathrm{x}=-0.6860$$

Commented by mrW1 last updated on 10/Jul/17

$${\mathrm{x}}^2=3^{\mathrm{x}}$$$$\Rightarrow \pm 3^{\frac{\mathrm{x}}{2}}=\mathrm{x}$$$$......$$$$\mathrm{totally}3\mathrm{solutions}$$

Commented by tawa tawa last updated on 10/Jul/17

$$\mathrm{ohh},\mathrm{i}\mathrm{see}....\mathrm{it}\mathrm{almost}\mathrm{affect}\mathrm{the}\mathrm{last}\mathrm{answer}$$

Commented by tawa tawa last updated on 10/Jul/17

$$\mathrm{Sir}....{\mathrm{x}}^3\mathrm{has}\mathrm{two}\mathrm{solutions}$$$${\mathrm{x}}^2\mathrm{has}\mathrm{three}\mathrm{solutions}...$$$$\mathrm{or}\mathrm{it}\mathrm{depends}\mathrm{on}\mathrm{the}\mathrm{question}.$$

Commented by tawa tawa last updated on 10/Jul/17

$$\mathrm{Show}\mathrm{me}\mathrm{sir}.$$

Commented by mrW1 last updated on 10/Jul/17

$$\mathrm{if}{\mathrm{x}}^3=3^{\mathrm{x}}$$$$\mathrm{since}{3}^{\mathrm{x}}>0$$$$\Rightarrow {\mathrm{x}}^3>0$$$$\Rightarrow \mathrm{x}>0$$$$\Rightarrow {\mathrm{x}}^3=3^{\mathrm{x}}\equiv \mathrm{x}=3^{\frac{\mathrm{x}}{3}}$$$$$$$$\mathrm{if}{\mathrm{x}}^2=3^{\mathrm{x}}$$$$\mathrm{x}\mathrm{can}\mathrm{also}\mathrm{be}\mathrm{negative},$$$$\Rightarrow {\mathrm{x}}^2=3^{\mathrm{x}}\equiv \mathrm{x}=\pm 3^{\frac{\mathrm{x}}{2}}$$$$$$$$\mathrm{just}\mathrm{like}$$$${\mathrm{x}}^3=8$$$$\Rightarrow \mathrm{x}={\left(8\right)}^{\frac{1}{3}}=2$$$${\mathrm{x}}^3=-8$$$$\Rightarrow \mathrm{x}={(-8)}^{\frac{1}{3}}=-2$$$$\mathrm{but}$$$${\mathrm{x}}^2=4$$$$\Rightarrow \mathrm{x}=\pm {\left(4\right)}^{\frac{1}{2}}=\pm 2$$

Commented by tawa tawa last updated on 10/Jul/17

$$\mathrm{Now},\mathrm{i}\mathrm{understand}\mathrm{sir}.\mathrm{God}\mathrm{bless}\mathrm{you}$$

Commented by mrW1 last updated on 10/Jul/17

$$\mathrm{solution}\mathrm{for}{\mathrm{x}}^4=3^{\mathrm{x}}$$$$\mathrm{x}=\pm 3^{\frac{\mathrm{x}}{4}}$$$$\mathrm{x}=\pm {\mathrm{e}}^{\frac{\mathrm{x}}{4}\ln 3}$$$${\mathrm{xe}}^{-\frac{\mathrm{x}}{4}\ln 3}=\pm 1$$$$(-\frac{\mathrm{x}}{4}\ln 3){\mathrm{e}}^{-\frac{\mathrm{x}}{4}\ln 3}=\pm \frac{\ln 3}{4}$$$$-\frac{\mathrm{x}}{4}\ln 3=\mathrm{W}(\pm \frac{\ln 3}{4})$$$$\Rightarrow \mathrm{x}=-\frac{4}{\ln 3}\mathrm{W}(\pm \frac{\ln 3}{4})$$$$\mathrm{x}=\{\begin{array}{l}-\frac{4}{\ln 3}\mathrm{W}\left(\frac{\ln 3}{4}\right)=-0.80225\\ -\frac{4}{\ln 3}\mathrm{W}(-\frac{\ln 3}{4})=\{\begin{array}{l}1.51687\\ 7.17476\end{array}\end{array}$$

Commented by tawa tawa last updated on 10/Jul/17

$$\mathrm{I}\mathrm{really}\mathrm{appreciate}\mathrm{your}\mathrm{effort}\mathrm{sir}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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