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Question Number 14853 by tawa tawa last updated on 04/Jun/17

Solve for x 3^(2x) = 18x

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x} \\ $$$$\mathrm{3}^{\mathrm{2x}} \:=\:\mathrm{18x} \\ $$

Answered by mrW1 last updated on 04/Jun/17

e^(2xln 3) =18x 18xe^(−2xln 3) =1 (−2xln 3)e^((−2xln 3)) =−((ln 3)/9) ⇒−2xln 3=W(−((ln 3)/9)) ⇒x=−((W(−((ln 3)/9)))/(2ln 3))≈−((W(−0.122068))/(2.197225)) = { ((−((−3.295837)/(2.197225))=1.5)),((−((−0.140479)/(2.197225))=0.063935)) :}

$${e}^{\mathrm{2}{x}\mathrm{ln}\:\mathrm{3}} =\mathrm{18}{x} \\ $$$$\mathrm{18}{xe}^{−\mathrm{2}{x}\mathrm{ln}\:\mathrm{3}} =\mathrm{1} \\ $$$$\left(−\mathrm{2}{x}\mathrm{ln}\:\mathrm{3}\right){e}^{\left(−\mathrm{2}{x}\mathrm{ln}\:\mathrm{3}\right)} =−\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{9}} \\ $$$$\Rightarrow−\mathrm{2}{x}\mathrm{ln}\:\mathrm{3}=\boldsymbol{{W}}\left(−\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{9}}\right) \\ $$$$\Rightarrow{x}=−\frac{\boldsymbol{{W}}\left(−\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{9}}\right)}{\mathrm{2ln}\:\mathrm{3}}\approx−\frac{{W}\left(−\mathrm{0}.\mathrm{122068}\right)}{\mathrm{2}.\mathrm{197225}} \\ $$$$=\begin{cases}{−\frac{−\mathrm{3}.\mathrm{295837}}{\mathrm{2}.\mathrm{197225}}=\mathrm{1}.\mathrm{5}}\\{−\frac{−\mathrm{0}.\mathrm{140479}}{\mathrm{2}.\mathrm{197225}}=\mathrm{0}.\mathrm{063935}}\end{cases} \\ $$

Commented by adelson last updated on 05/Jun/17

what′s W ?

$${what}'{s}\:\boldsymbol{{W}}\:? \\ $$

Commented by tawa tawa last updated on 05/Jun/17

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\: \\ $$

Commented by mrW1 last updated on 05/Jun/17

W stands for Lambert W−function which is the inverse function of f(x)=xe^x

$${W}\:{stands}\:{for}\:{Lambert}\:{W}−{function} \\ $$$${which}\:{is}\:{the}\:{inverse}\:{function}\:{of} \\ $$$${f}\left({x}\right)={xe}^{{x}} \\ $$

Commented by tawa tawa last updated on 05/Jun/17

Have been wondering, please sir explain further.

$$\mathrm{Have}\:\mathrm{been}\:\mathrm{wondering},\:\mathrm{please}\:\mathrm{sir}\:\mathrm{explain}\:\mathrm{further}. \\ $$

Commented by mrW1 last updated on 05/Jun/17

I don′t know what I can tell more. One should know that for real values W(x) function is defined for x≥−(1/e) and for −(1/e)<x<0 there are two values for W(x).

$${I}\:{don}'{t}\:{know}\:{what}\:{I}\:{can}\:{tell}\:{more}. \\ $$$${One}\:{should}\:{know}\:{that}\:{for}\:{real}\:{values} \\ $$$${W}\left({x}\right)\:{function}\:{is}\:{defined}\:{for}\:{x}\geqslant−\frac{\mathrm{1}}{{e}} \\ $$$${and}\:{for}\:−\frac{\mathrm{1}}{{e}}<{x}<\mathrm{0}\:{there}\:{are}\:{two}\:{values} \\ $$$${for}\:{W}\left({x}\right). \\ $$

Commented by tawa tawa last updated on 05/Jun/17

Alright sir. God bless you

$$\mathrm{Alright}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you} \\ $$

Commented by tawa tawa last updated on 05/Jun/17

What i mean is that how do you get − 3.295837

$$\mathrm{What}\:\mathrm{i}\:\mathrm{mean}\:\mathrm{is}\:\mathrm{that}\:\mathrm{how}\:\mathrm{do}\:\mathrm{you}\:\mathrm{get}\:\:−\:\mathrm{3}.\mathrm{295837} \\ $$

Commented by mrW1 last updated on 06/Jun/17

if you can not find a W−function calculator which gives 2 values for x<0, you can also calculate by youself, for example using geogebra. you find just the zero points from function f(x)=xe^x +((ln 3)/9)

$${if}\:{you}\:{can}\:{not}\:{find}\:{a}\:{W}−{function} \\ $$$${calculator}\:{which}\:{gives}\:\mathrm{2}\:{values}\:{for} \\ $$$${x}<\mathrm{0},\:{you}\:{can}\:{also}\:{calculate}\:{by}\:{youself}, \\ $$$${for}\:{example}\:{using}\:{geogebra}.\:{you} \\ $$$${find}\:{just}\:{the}\:{zero}\:{points}\:{from}\:{function} \\ $$$${f}\left({x}\right)={xe}^{{x}} +\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{9}} \\ $$

Commented by mrW1 last updated on 06/Jun/17

Commented by tawa tawa last updated on 06/Jun/17

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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