Question

Does anyone know how to solve this equation: y+ln(y)=10

Answer 1

Doing some simple algebra we get
\[\ln (e ^{y})+\ln(y)=10\]
\[\ln (ye^{y})=10\]
\[ye^{y}=e^{10}\]
To solve this we need to know the Lambert W-Function which is the inverse function of
\[f(W)=We^{W}\]
(very good article about this here http://mathworld.wolfram.com/LambertW-Function.html)
So solution of this equation looks like this:
\[y=W(e^{10})\]
The Lambert W-function cannot be expressed in terms of elementary functions but we can get the numeriacal approximation (because there is the Taylor series for this function):
\[W(e^{10}) \approx 7.92942...\]
( http://www.wolframalpha.com/input/?i=W%28e%5E10%29)
Check our approximate solution:
\[7.92942 + \ln (7.92942) = 9.999999892997119 \approx 10\]

Answer 2

As you are obliged to use numerical solutions, fixed point is easy to use, even with a simple calculator, and it is recommanded as for x>1, |g'(x)|<1 where g(x)=10 - ln(x).

Answer 3

Yes, I should have been more specific in my question. I was trying to find out if there was an analytical solution, apparently there isn't. I can write a fairly simple matlab program to solve it. Really, I'm interested in an analytical solution because the problem that I'm working with is much more complex, but it has the same form as the stated question. Anyway, I've looked around and I don't think that there is an analytical solution.