# Solving x^x = a*x with Lambert W Function

• Swapnil
In summary, the conversation discusses the method of using the Lambert W function to solve the equation x^x=ax, but it is ultimately determined that this method is not suitable. The conversation then explores a method using complex numbers and equations to find solutions, but it is concluded that there is no closed-form solution for this equation and numerical methods must be used.
Swapnil
How would you solve
$$x^x = ax$$
for x (where a is any postive constant >1)?

I am trying the Lambert W function but I just can't get it to the right form. I am starting to think that you can't solve this using the Lambert W function. Any help?

edit: Sorry, I really wanted the expression to be x^x = a*x

Last edited:
You're looking for solutions to $x^x=cx$.

Set x=a+ib=|x|exp(i$\Phi$) and look to solutions to

$$|x|\exp(i\Phi)^{a+ib}=|x|\exp(i\Phi)$$

knowing that two complex numbers are equal iff their modulus are equal and their phase differ by at most a factor of 2n$\pi$, $n\in \mathbb{Z}$. For instance, I find that the condition of equality of modulus imposes the following relation btw a,b and c

$$\frac{b-ab}{2}\ln(a^2+b^2)\tan^{-1}(b/a)=c$$

The condition on the phase will restrict the possible solutions some more.

Last edited:
I have a little doubt about what I wrote though because I don't remember how a complex number raised to a complex number "looks" like but I think

$$(e^{z})^w=e^{wz}$$

is correct.

quasar987 said:
$$\frac{b-ab}{2}\ln(a^2+b^2)\tan^{-1}(b/a)=c$$

The condition on the phase will restrict the possible solutions some more.
I don't know how is this going to help. Instead of having one equation and one unknown we would now have two equations and two unknowns. And it doesn't seem like this approach is going to be any easy looking how the a's and b's are "trapped" inside.

Wheter you find it pretty or not it's the solution nonetheless. If you give that to a computer, he will indiscriminately find all solutions in a fraction of second.

Last edited:
quasar987 said:
Wheter you find it pretty or not it's the solution nonetheless. If you give that to a computer, he will indiscriminatorily find all solutions in a fraction of second.
That's the only clever way to go here.
Closed form solution is impossible for this equation I think.Just numerically ,and it depends on constant $$a$$.

So it is NOT possible to use the Lambert W function here. I mean, we can use the Lambert W function to find a closed-form solution to the equation x^x = a. But we can't, supposedly, solve for x^x=ax using the same function?

Anyways, does anyone have clever way to solve this equation in CLOSED-FORM?

Possible or not,this equation can't be solved explicitly.I haven't worked with "W" function much in past ,but I think you are right:
X^(X)=a is one thing and X^(X-1)=a quite another one.
There are numerical methods though.
Some may be better suited for this type of equation (ie. faster convergence) than others. It depends what precision you want.

## 1. What is the Lambert W function?

The Lambert W function, also known as the omega function, is a special function that is used to solve equations of the form x^x = a*x. It is defined as the inverse function of f(x) = x*e^x, and is denoted as W(x).

## 2. How is the Lambert W function used to solve equations?

The Lambert W function is used to solve equations in which the variable appears in both the base and exponent, such as x^x = a*x. By applying the function to both sides of the equation, the variable can be isolated and solved for.

## 3. Can the Lambert W function be used for all values of a?

No, the Lambert W function can only be used for certain values of a. Specifically, it can be used when a*e^a is greater than or equal to -1/e. This is known as the branch point of the function.

## 4. How many solutions can the Lambert W function provide for an equation?

The Lambert W function can provide two solutions for an equation of the form x^x = a*x. However, one of the solutions may not be real, depending on the value of a. This is known as the principal branch of the function.

## 5. Are there any other applications of the Lambert W function?

Yes, the Lambert W function has various applications in mathematical and scientific fields, such as in population growth models, physics equations, and financial calculations. It also has connections to other special functions, such as the gamma function and the zeta function.

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