Thread: How to find the solution 8*x/e^x = 2

1. How to find the solution 8*x/e^x = 2

With a little bit of algebra I can also write this equation as ln 4*x = x and 4*x = e^x but that doesn't help anything.
I'm wondering wether it is possible to find the solution of this equation without using a graphing calculator and without plugging in numbers in a methodical way.

2. You won't be able to solve that in terms of "elementary functions". You could, however, rewrite it as $\displaystyle xe^{-x}= 2$ so that $\displaystyle -xe^{-x}= -2$ and then, letting y= -x, $\displaystyle ye^y= -2$.

Now, use "Lambert's W function" which is defined as "the inverse function to $\displaystyle f(x)= xe^{x}$". Taking W of both sides, W(ye^y)= y= W(-2). Since y= -x, -x= W(-2) and x= -W(-2).

3. You have discovered a great oddity. Such a simple equation that has no "algebraic" solution. You must use numerical methods. There is no "closed-form" solution for such things unless you invent one. q.v. Lambert's W function.

There are many numerical methods that will give you as precise a result as desired - some more quickly than others.

4. Suppose the problem had been $\displaystyle e^x= 2$. Would you say the same thing- "You must use numerical methods. There is no "closed-form" solution for such things unless you invent one. q.v. the natural logarithm"?

If not, how are these two problems different?