What are the solutions for 10 = (e^x)/x?

[Tivet]

Hey, new here, I just want to know how can i solve that, i mean i know that there is 2 solutions and i can pove it pretty easily but. How can i found solution x₁ and x₂ for the following equation :

10=e^x/x

Hope anyone can help >.< ( PS: I'm foreign please forgive me about my orthography)

 

[fresh_42]

Hello, and

What do you regard as a solution? It can be expressed with the Lambert W-function or given numerically.

 

[Tivet]

Well in my assignment i need to express it numerically, which i can do really fine. But i wanted to go further and try to express it with the ProductLog function but dindn't found any ways to make it like x = ... with no x in the " ... " >.<

 

[fresh_42]

Well in my assignment i need to express it numerically, which i can do really fine. But i wanted to go further and try to express it with the ProductLog function but dindn't found any ways to make it like x = ... with no x in the " ... " >.<

You won't find any, because there is none. All you can achieve is an expression with the Lambert W-function like the one here. (And this function isn't really one.)

Edit: And of course the series expansion.

 

[Tivet]

Sweet, thanks for the help

 

[Tivet]

Wait hold on what does the -1 on the W-1(-0.1) means ?

 

[fresh_42]

Have a look on the Wikipedia page I linked to:

The additional constraint $W ≥ −1$ defines a single-valued function $W_0(x)$. We have $W_0(0) = 0$ and $W_0(−1/e) = −1$. Meanwhile, the lower branch has $W ≤ −1$ and is denoted $W_{−1}(x).$It decreases from $W_{−1}(−1/e) = −1$ to $W_{−1}(0^−) = −∞.$