Given the following two functions, how can I determine an equation for when the first is greater than the second?
$$n\log(k)$$ and $$n + k\log(n)$$
I graphed them, and found a distinct intersection line of the planes, but mathematica was unable to solve the intersection.
My intuition says to take partial derivatives of both, but I am unsure how to use the information from those derivatives to find the equation.
Clearly you have $k$ and $n$ both positive to have real logarithms
I suspect you can separate the variables to get $$\frac{\log(k) -1}{k} = \frac{\log(n)}{n}$$
A solution for this will involve the Lambert W function or something similarly complicated and I think will give $$n = e^{-W\left(-\frac{\log(k)-1}{k}\right)}$$
Note that the Lambert W function is multivalued and that for $k \gt e$ there will be two values of $n$ satisfying the equality
So for example with $k=10$, you get $n\log(k) \gt n + k\log(n)$ when either $0 \lt n \lt 1.16366878\ldots$ or $n \gt 24.5819976\ldots$