Finding all the solutions of e^z=ln(z)

Hello,

I was looking bprp solving e^x=ln(x) (here's the link https://www.youtube.com/watch?v=EMu-kYY5rdE).

I realize, for real values of logarithm and exponential, I can translate both ln(x) and e^x, respectively up and down, by the same value, to get exactely one real intersection. Then, for any bigger k, you get 2 intersections.

So, I started to solve e^x-k=ln+k, x and k are real and k>0, but I couldn't express a solution (for one real intersection, it seems that k is approximately 1.15). Thus, I can't solve the general case.

Does someone have an idea?

Bye,

Nicolas