Solution to equation

chicodesouza

Is there any algebraic solution to the equation:
$$\displaystyle \frac{e^x-e^0}{x-0}=50 ?$$

Regards

frick

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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