Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 175834 by Linton last updated on 08/Sep/22

xe^x^(1/)  = e  solve for x

$${xe}^{\overset{\mathrm{1}/} {{x}}} =\:{e} \\ $$$${solve}\:{for}\:{x} \\ $$

Commented by mr W last updated on 08/Sep/22

put your question in order!  do you mean xe^(1/x) =e?  then it′s clear x=1.

$${put}\:{your}\:{question}\:{in}\:{order}! \\ $$$${do}\:{you}\:{mean}\:{xe}^{\frac{\mathrm{1}}{{x}}} ={e}? \\ $$$${then}\:{it}'{s}\:{clear}\:{x}=\mathrm{1}. \\ $$

Commented by Linton last updated on 08/Sep/22

please show working

$${please}\:{show}\:{working} \\ $$

Commented by mr W last updated on 08/Sep/22

if the question is xe^(1/x) =e, then it′s  obvious that x=1.  for xe^(1/x) =3, the solution is not obvious,  but it can be solved using lambert W  function, see below.

$${if}\:{the}\:{question}\:{is}\:{xe}^{\frac{\mathrm{1}}{{x}}} ={e},\:{then}\:{it}'{s} \\ $$$${obvious}\:{that}\:{x}=\mathrm{1}. \\ $$$${for}\:{xe}^{\frac{\mathrm{1}}{{x}}} =\mathrm{3},\:{the}\:{solution}\:{is}\:{not}\:{obvious}, \\ $$$${but}\:{it}\:{can}\:{be}\:{solved}\:{using}\:{lambert}\:{W} \\ $$$${function},\:{see}\:{below}. \\ $$

Commented by mr W last updated on 08/Sep/22

xe^(1/x) =3  (1/x)e^(−(1/x)) =(1/3)  (−(1/x))e^(−(1/x)) =−(1/3)  −(1/x)=W(−(1/3))  ⇒x=−(1/(W(−(1/3))))  =−(1/(−1.512135 or −0.619061))= { ((0.661317)),((1.615349)) :}

$${xe}^{\frac{\mathrm{1}}{{x}}} =\mathrm{3} \\ $$$$\frac{\mathrm{1}}{{x}}{e}^{−\frac{\mathrm{1}}{{x}}} =\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\left(−\frac{\mathrm{1}}{{x}}\right){e}^{−\frac{\mathrm{1}}{{x}}} =−\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$−\frac{\mathrm{1}}{{x}}={W}\left(−\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$$$\Rightarrow{x}=−\frac{\mathrm{1}}{{W}\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)} \\ $$$$=−\frac{\mathrm{1}}{−\mathrm{1}.\mathrm{512135}\:{or}\:−\mathrm{0}.\mathrm{619061}}=\begin{cases}{\mathrm{0}.\mathrm{661317}}\\{\mathrm{1}.\mathrm{615349}}\end{cases} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com