Logarithms in Systems of Equations

[Mandelbroth]

Homework Statement

Consider the system...

$y=e^{-x}+1 \\ y=2+lnx$

The Attempt at a Solution

I don't think there is a "pretty way" (algebraic manipulation) to do this. I simply found the point where the plots intercepted, getting the solution set (0.62745018..., 1.5339409...).

Is there a better way to do this? For what it's worth, this was for a friend who claims that her math teacher is unreasonable, so I don't know if it does has a fully algebraic approach.

Edit: For clarity, both have a domain and codomain in the set of real numbers, so there isn't an imaginary part to worry about.

 

[mfb]

Is there a better way to do this?

In general, there is not. Some equations have an analytic solution which is easy to find (x+e^x = 1 for example), but the general case does not have a nice solution.