Solution to equation

Jun 2014
1
0
Brazil
Is there any algebraic solution to the equation:
\(\displaystyle
\frac{e^x-e^0}{x-0}=50 ?
\)

Regards
 
May 2019
7
7
Kansas
You are asking about \(\displaystyle e^x= 50x\). Rewrite it as \(\displaystyle xe^{-x}= \frac{1}{50}\). Let y= -x. Then x= -y and the equation becomes \(\displaystyle -ye^y= \frac{1}{50}\) or \(\displaystyle ye^y= -\frac{1}{50}\). Now apply the Lambert W function (which is defined as the inverse function to \(\displaystyle f(x)= xe^x\)) to both sides: \(\displaystyle y= W\left(-\frac{1}{50}\right)\) so \(\displaystyle x= -W\left(-\frac{1}{50}\right)\).
 
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