Is there an exact solution for (7-x)ln(x)?

[dav2008]

This is part of an optimization problem, and I'm finding the zeros of the first derivative which is f'(x) = (7-x)/x - ln(x)

Is there any way to solve this for an exact answer? I've tried and there doesn't seem to be a way. Is it even possible to solve for an exact answer?

0=(7-x)/x - ln(x)

Thanks.

 

[HallsofIvy]

The only "exact" solution for an equation that involves x both in and outside a logarithm involves Lambert's W function (which is defined as the inverse to f(x)= xe^x). Other than that you would need to use numerical methods.

Do you have any reason to think there should be an "exact" solution?
What was the orginal function to be optimised?

(7-x)/x - ln(x) is, of course, the derivative of (7-x)ln(x). Was that your function?

 

[dav2008]

Originally posted by HallsofIvy
The only "exact" solution for an equation that involves x both in and outside a logarithm involves Lambert's W function (which is defined as the inverse to f(x)= xe^x). Other than that you would need to use numerical methods.

Do you have any reason to think there should be an "exact" solution?
What was the orginal function to be optimised?

(7-x)/x - ln(x) is, of course, the derivative of (7-x)ln(x). Was that your function?

Yeah it was (7-x)ln(x). It could have also been (7-e^y)(y) but eventually it is pretty much the same thing.

The only reason I'm looking for an exact answer is because the homework assignment asks for one. I was just making sure that it really wasn't possible. Tx.