Question and Answers Forum

All Questions      Topic List

Arithmetic Questions

Previous in All Question      Next in All Question      

Previous in Arithmetic      Next in Arithmetic      

Question Number 66185 by Tony Lin last updated on 10/Aug/19

$${\left(e^{\frac{1}{e}}\right)}^{{\left(e^{\frac{1}{e}}\right)}^{.{\cdot }^{.\left(e^{\frac{1}{e}}\right)}}}=?$$

Commented by GordonYeeman last updated on 10/Aug/19

$${\left(e^{\frac{1}{e}}\right)}^x=x$$$$\frac{x}{e}lne=lnx$$$$\frac{lne}{e}=\frac{lnx}{x}\to x=e$$

Answered by mr W last updated on 10/Aug/19

$${\left(e^{\frac{1}{e}}\right)}^{{\left(e^{\frac{1}{e}}\right)}^{.{\cdot }^{.\left(e^{\frac{1}{e}}\right)}}}=x$$$$\Rightarrow {\left(e^{\frac{1}{e}}\right)}^x=x$$$$\Rightarrow e^{\frac{x}{e}}=x$$$$\Rightarrow x=e$$$$$$$$or$$$$\Rightarrow {xe}^{-\frac{x}{e}}=1$$$$\Rightarrow (-\frac{x}{e})e^{-\frac{x}{e}}=-\frac{1}{e}$$$$\Rightarrow -\frac{x}{e}=W(-\frac{1}{e})=-1$$$$\Rightarrow x=e$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com