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Question Number 71016 by mr W last updated on 10/Oct/19

ln (e+ln (e+ln (e+...)))=?

$$\boldsymbol{\mathrm{ln}}\:\left(\boldsymbol{{e}}+\boldsymbol{\mathrm{ln}}\:\left(\boldsymbol{{e}}+\boldsymbol{\mathrm{ln}}\:\left(\boldsymbol{{e}}+...\right)\right)\right)=? \\ $$

Answered by mr W last updated on 11/Oct/19

ln (e+ln (e+ln (e+...)))=x ln (e+x)=x e+x=e^x (e+x)=e^(−e) e^(e+x) −(e+x)e^(−(e+x)) =−e^(−e) −(e+x)=W(−e^(−e) ) ⇒x=−[e+W(−e^(−e) )]= { ((−(e−0.070832)=−2.64745)),((−(e−4.138652)=1.42037)) :} but i think x>1, since x=ln (e+...)>ln e=1

$$\boldsymbol{\mathrm{ln}}\:\left(\boldsymbol{{e}}+\boldsymbol{\mathrm{ln}}\:\left(\boldsymbol{{e}}+\boldsymbol{\mathrm{ln}}\:\left(\boldsymbol{{e}}+...\right)\right)\right)={x} \\ $$$$\boldsymbol{\mathrm{ln}}\:\left(\boldsymbol{{e}}+{x}\right)={x} \\ $$$${e}+{x}={e}^{{x}} \\ $$$$\left({e}+{x}\right)={e}^{−{e}} {e}^{{e}+{x}} \\ $$$$−\left({e}+{x}\right){e}^{−\left({e}+{x}\right)} =−{e}^{−{e}} \\ $$$$−\left({e}+{x}\right)={W}\left(−{e}^{−{e}} \right) \\ $$$$\Rightarrow{x}=−\left[{e}+{W}\left(−{e}^{−{e}} \right)\right]=\begin{cases}{−\left({e}−\mathrm{0}.\mathrm{070832}\right)=−\mathrm{2}.\mathrm{64745}}\\{−\left({e}−\mathrm{4}.\mathrm{138652}\right)=\mathrm{1}.\mathrm{42037}}\end{cases} \\ $$$${but}\:{i}\:{think}\:{x}>\mathrm{1},\:{since}\:{x}=\mathrm{ln}\:\left({e}+...\right)>\mathrm{ln}\:{e}=\mathrm{1} \\ $$

Answered by MJS last updated on 10/Oct/19

ln (e+x) =x approximating I get x=−2.64745∨x=1.42037

$$\mathrm{ln}\:\left({e}+{x}\right)\:={x} \\ $$$$\mathrm{approximating}\:\mathrm{I}\:\mathrm{get} \\ $$$${x}=−\mathrm{2}.\mathrm{64745}\vee{x}=\mathrm{1}.\mathrm{42037} \\ $$

Commented by mr W last updated on 11/Oct/19

thanks sir! how to explain that two values are possible for the limit?

$${thanks}\:{sir}! \\ $$$${how}\:{to}\:{explain}\:{that}\:{two}\:{values}\:{are} \\ $$$${possible}\:{for}\:{the}\:{limit}? \\ $$

Commented by MJS last updated on 11/Oct/19

that′s interesting. only the positive value is valid, the negative value comes in the same way as false solutions appear when we square an equation.

$$\mathrm{that}'\mathrm{s}\:\mathrm{interesting}.\:\mathrm{only}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{value}\:\mathrm{is} \\ $$$$\mathrm{valid},\:\mathrm{the}\:\mathrm{negative}\:\mathrm{value}\:\mathrm{comes}\:\mathrm{in}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{way}\:\mathrm{as}\:\mathrm{false}\:\mathrm{solutions}\:\mathrm{appear}\:\mathrm{when}\:\mathrm{we}\:\mathrm{square} \\ $$$$\mathrm{an}\:\mathrm{equation}. \\ $$

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