Say I have the function $y=4e^{-2x}-3x$. I can use a graphing calculator to approximately determine the roots, but how do I find an exact answer?
2026-04-05 22:17:58.1775427478
Solving a binomial when one of the terms is in the form $e^x$
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An exact answer is only possible in terms of the Lambert $W$ function (Wikipedia link): $$\begin{align*} 0 &= 4e^{-2x}-3x\\\\ 3x &= 4e^{-2x}\\\\ 2xe^{2x}&=\tfrac{8}{3}\\\\ ye^y&=\tfrac{8}{3}\;\;(y=2x)\\\\ y&=W(\tfrac{8}{3})\\\\ x&=\tfrac{1}{2} W(\tfrac{8}{3}) \end{align*}$$