# What is the solution of x*exp(-x)=1?

1. Oct 11, 2007

### Odyssey

1. The problem statement, all variables and given/known data
What is the solution of the equation x*exp(-x)=1?

2. Relevant equations

3. The attempt at a solution
I tried taking the ln of both sides...and I got
ln(x*exp(-x))=ln(1)
ln(x)-ln(exp(x))=ln(1)
ln(x)-x=ln(1)

Then I don't know what to do next...am I on the right track at least?

Thanks for the help! :)

2. Oct 11, 2007

### futurebird

$$xe^{-x}=1$$

Use what you know about negative exponents to rewrite this problem in a more simple form. Where is this problem from? You may need to show that there is no solution rather than finding the solution.

3. Oct 11, 2007

### Dick

futurebird is right. You may have to show it doesn't have a solution. Clearly x>0 is the only possibility. So you got ln(x)-x=ln(1)=0. Take f(x)=ln(x)-x. Look at f'(x) to figure out where it has a max/min etc. Can it ever be zero?

4. Oct 12, 2007

### christianjb

x=1 works.

5. Oct 12, 2007

### futurebird

You scared me for a moment there!

An easy way to see that there is so solution is to first simplify then graph both sides of the equation by setting them equal to y.

6. Oct 12, 2007

### Gib Z

Graphing them both and showing the don't intersect is not really the most rigorous proof...

Use Dick's method in post 3.

7. Oct 12, 2007

### Odyssey

ahhhh...thanks guys! Why didnt I think of that? =\