Solve the given equation using the Lambert W function

[chwala]

Homework Statement

Solve for $x$ given $3^x=2x+2$

Relevant Equations

Lambert W Function

I just came across this...the beginning steps are pretty easy to follow...i need help on the highlighted part as indicated below;

From my own understanding, allow me to create my own question for insight purposes...

let us have;

$7^x=5x+5$

$\dfrac{1}{5}=(x+1)7^{-x}$

$\dfrac{1}{35}=(x+1)7^{(-x-1)}$ this is clear...
then we desire our equation to be in the form;
$we^w$
then we shall have,

$-\dfrac{1}{35}=(-x-1)7^{(-x-1)}$

Let $y=7^{(-x-1)}$

then $\ln y=(-x-1)\ln 7$

$⇒e^{(-x-1)\ln 7}=y$ then on substituting back on $-\dfrac{1}{35}=(-x-1)7^{(-x-1)}$ and multiplying both sides of the equation by $\ln 7$

we get;

$(\ln 7)(-x-1)⋅ e^{(-x-1)\ln 7} = \dfrac {-\ln 7}{35}$

$(\ln 7)(-x-1)=W_0 \left[\frac{-\ln 7}{35}\right ]$

or
$(\ln 7)(-x-1)=W_{-1} \left[\frac{-\ln 7}{35}\right ]$ how do we arrive at the required values from here? Is there a table?

 

[nuuskur]

The values of Lambert W are numerically evaluated. I believe Newton's method should be a solid pick and wikipedia recommends it, as well. Otherwise, it looks like you have correctly followed the guide. I wouldn't worry too much about numerical values unless you needed to implement them.

 

[chwala]

I hear you. Thanks for your feedback. I may explore this in detail later on how they arrived at the numerical values. Newton's method handles this quite well.