Lambert W Function

Definition

The Lambert W function (also called the omega function or product log) is the inverse function of:

$f(w)=w{e}^w$

So, for any complex number $z$,

$z=W(z)e^{W(z)}$

Beautiful Example

Problem: Solve for $x$ in $4^x+x=260$.

1. Isolate the exponential

$260-x=4^x⟹(260-x)\cdot 4^{-x}=1$

2. Transform to $w{e}^w$ form

We use the identity $a^b=e^{b\ln a}$ and manipulate the constants:

$(260-x)\cdot e^{-x\ln 4}=1$

Multiply both sides by $4^{260}$ to align the exponent with the linear term:

$(260-x)\cdot e^{(260-x)\ln 4}=4^{260}$

Multiply by $\ln 4$ to complete the $w{e}^w$ pattern:

$\underset{w}{\underbrace{(260-x)\ln 4}}\cdot e^{\stackrel{w}{\overbrace{(260-x)\ln 4}}}=4^{260}\ln 4$

3. Apply the Lambert W Function

$(260-x)\ln 4=W(4^{260}\ln 4)$

$x=260-\frac{W(4^{260}\ln 4)}{\ln 4}$

4. The Elegant Simplification

Recognizing that $4^4=256$, we can rewrite the argument of $W$:

$W(4^{260}\ln 4)=W(4^{256}\cdot 4^4\cdot \ln 4)$

$W(4^{260}\ln 4)=W(256\ln 4\cdot e^{256\ln 4})$

Since $W(y{e}^y)=y$:

$W(256\ln 4\cdot e^{256\ln 4})=256\ln 4$

Final Answer

$x=260-\frac{256\ln 4}{\ln 4}=260-256=4$