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

[Odyssey]

Homework Statement

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

Homework Equations

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! :)

 

[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.

 

[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?

 

[christianjb]

x=1 works.

Ooh- my bad- misread the question. It doesn't have a solution.

 

[futurebird]

x=1 works.

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.

 

[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.

 

[Odyssey]

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