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August 13th, 2019, 04:42 AM   #1
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how can I solve x/ln(x)=n

Hi, I am working on some program and the answer for this education is very important. Should I use Lambert W-function and if I do then how?
Can any one send the steps for solving this problem, for example?
x/ln(x)=100
Okay, the answer to this can help me as well:
10t+e^t=100

Last edited by skipjack; August 13th, 2019 at 09:09 AM.
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August 13th, 2019, 06:30 AM   #2
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Apply Lambert W-function.
(1) $\displaystyle \: W_{0} \left(-\frac{\ln x}{x}\right)=-\ln x$ for $x<e$.
(2) $\displaystyle \: p^x =ax +b $ has solution $\displaystyle -(\ln p)^{-1} W\left(-a^{-1} \ln p \cdot p^{-b/a}\right)-\frac{b}{a}$.

Last edited by skipjack; August 13th, 2019 at 08:15 AM.
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August 13th, 2019, 09:05 AM   #3
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For the first problem, $x = e^{-W_0(-1/100)}$ or $e^{-W_{-1}(-1/100)}$.

For the second problem, the real solution is $\displaystyle t = 10 - W\!\left(\frac{e^{10}}{10}\right)\!$.
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