# Searching for a solution for a long time

Hello all, I'm new to the forums (signed up to ask this question- this isn't a homework question). My friend gave me this equation a while back and all of my math teachers that I've asked have been stumped. I'm currently a sophomore in high school with no calculus experience (currently Algebra 2 honors), so this problem threw me off. If you can find a solution, it'd be great if you could explain to me how you came about it in simple (if possible) explanation.

The problem:

x(e^x) = 3
Solve for x

Thanks a lot guys!

Last edited:

phyzguy
What do you mean by "solve"? Do you just want to know the value of x? If so you can find the answer graphically by plotting the function e^x and the function 3/x on the same graph. The solution will be where they intersect, and it has a value of about 1.05. If you want to write x in terms of "elementary functions" (sines, cosines, polynomials, etc), that is not possible. However, there is a defined function that will give x. It is called the "Lambert W" function or the "Product Log" function. Here is the Wikipedia page on it.

If you want to write x in terms of "elementary functions" (sines, cosines, polynomials, etc), that is not possible.

Does somebody on PF know how to prove this?

Chestermiller
Mentor
Re-express the problem in the form:

x = ln (3/x)

You can solve this equation by the method of successive substitutions. This is an iterative method, where you have a starting guess, x = x0, and then you calculate successive values of x (on the left side of the equation) using previous values of x (on the right side of the equation). The successive substitution equation is:

$$x_{n+1}=ln \left(\frac{3}{x_n}\right)$$

You keep repeating the calculation until x stops changing. Try doing this with an initial guess for x of x0 = 1. You can do this on a spreadsheet or on a hand calculator.

Another iterative method that can be used is the half-interval method (aka bisection method). Look this up on Google.

Chet

Only a little motivational question: I happen to have an AMS book Algebraic Groups and Differential Galois Theory By Crepo and Hajto on my shelf, but I haven't read it almost at all. Do you think that reading it will help in this direction?

Re-express the problem in the form:

x = ln (3/x)

You can solve this equation by the method of successive substitutions. This is an iterative method, where you have a starting guess, x = x0, and then you calculate successive values of x (on the left side of the equation) using previous values of x (on the right side of the equation). The successive substitution equation is:

$$x_{n+1}=ln \left(\frac{3}{x_n}\right)$$

Writing fixed point iterations like this is dice game, because it is difficult to tell how convergent or divergent they are going to be. For example, your reply certainly left open the question why not write the problem in form

$$x = ae^{-x}$$

and start iterating

$$x_{n+1} = ae^{-x_n}$$

Chestermiller
Mentor
Only a little motivational question: I happen to have an AMS book Algebraic Groups and Differential Galois Theory By Crepo and Hajto on my shelf, but I haven't read it almost at all. Do you think that reading it will help in this direction?

Writing fixed point iterations like this is dice game, because it is difficult to tell how convergent or divergent they are going to be. For example, your reply certainly left open the question why not write the problem in form

$$x = ae^{-x}$$

and start iterating

$$x_{n+1} = ae^{-x_n}$$
True. It helps to have some experience. But there is a of determining whether the successive substitution interation is going to be convergent or not.

Chet