# Solve the given equation using the Lambert W function

• chwala
In summary, the conversation discusses the solution process for a given equation, with specific attention to transforming the equation into the form of ##we^w## and using the values of Lambert W. The use of Newton's method is recommended for evaluating the numerical values of Lambert W.
chwala
Gold Member
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?

Last edited:
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.

jim mcnamara
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.

## What is the Lambert W function?

The Lambert W function, also known as the omega function, is a special function that is defined as the inverse of the function f(z) = ze^z. It has many applications in mathematics, physics, and engineering, and is particularly useful in solving equations involving exponential and logarithmic terms.

## How do you use the Lambert W function to solve equations?

To use the Lambert W function to solve an equation, you first need to rearrange the equation so that it is in the form f(z) = ze^z. Then, you can apply the Lambert W function to both sides of the equation to isolate the variable. Finally, you can solve for the variable using algebraic manipulation and the properties of the Lambert W function.

## What types of equations can be solved using the Lambert W function?

The Lambert W function can be used to solve a variety of equations, including exponential and logarithmic equations, as well as equations that involve both exponential and logarithmic terms. It is particularly useful in solving equations that cannot be solved using traditional algebraic methods.

## Are there any limitations to using the Lambert W function?

While the Lambert W function is a powerful tool for solving equations, it does have some limitations. It can only be used to solve equations that can be rearranged into the form f(z) = ze^z. Additionally, the solutions obtained using the Lambert W function may not always be real numbers, and may require further simplification or approximation.

## What are some real-world applications of the Lambert W function?

The Lambert W function has many applications in mathematics, physics, and engineering. It is commonly used in the fields of quantum mechanics, control theory, and population dynamics. It is also used in financial modeling and in solving differential equations that arise in various scientific and engineering problems.

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