Too transcendental for me

Calculus Level 3

What is the value of x ∈ R x \in \mathbb{R} when

x + e x = 2 ? x + e^x = 2?

OBS: W W denotes the Lambert W function.

sin ⁡ − 1 ( π 8 ) + cos ⁡ − 1 ( π 8 ) \sin^{-1}\left({\frac{\pi}{8}}\right) + \cos^{-1}\left({\frac{\pi}{8}}\right) tan ⁡ − 1 ( 2 ) \tan^{-1}(\sqrt{2}) 2 − W ( e 2 ) 2 - W(e^2) 1 2 W ( 1 2 ) \frac{1}{2} W\left(\frac{1}{\sqrt{2}}\right)

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1 solution

Joseph Newton
Jul 15, 2019

The Lambert W function is defined as the inverse of f ( x ) = x e x f(x)=xe^x . This means that W ( x e x ) = x W(xe^x)=x , so we simply have to rearrange the expression until it is in this form: x + e x = 2 e x = 2 − x 1 = ( 2 − x ) e − x e 2 = ( 2 − x ) e 2 − x W ( e 2 ) = W ( ( 2 − x ) e ( 2 − x ) ) W ( e 2 ) = 2 − x x = 2 − W ( e 2 ) \begin{aligned}x+e^x&=2\\ e^x&=2-x\\ 1&=(2-x)e^{-x}\\ e^2&=(2-x)e^{2-x}\\ W(e^2)&=W\left((2-x)e^{(2-x)}\right)\\ W(e^2)&=2-x\\ x&=2-W(e^2)\end{aligned} So, x = 2 − W ( e 2 ) x=\boxed{2-W(e^2)}

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