How to solve an exponential and logarithmic system of equations? $$
\left\{\begin{array}{c}
e^{2x} + e^y = 800 
\\ 3\ln(x) + \ln(y) = 5 
\end{array}\right.$$
I understand how to solve system of equations, logarithmic rules, and the fact that $\ln(e^x) = e^{\ln(x)} = x$. However, any direction I seem to go with this problem causes me difficulties. For example, when solving for y, I can get $y = \ln\left(800 - e^{2x}\right)$ or $y = \frac{e^5}{x^3}$. Both directions seem to lead me into a dead end. Although system of equations seems to be a frequent question on here, I haven't found anything helpful towards this problem.

So here is another solution for your problem, again using Newton's method we get by using as initial value $x_0=3.2$ (it is very sensitive!) the following line of iterates
$$x_0=3.2$$ 
$$x_1=3.330187$$ 
$$x_2=3.303383$$ 
$$x_3=3.302208$$ 
$$x_4=3.302206$$
which gives us as an approximate solution
$$
f(3.302206)\approx-5.277731 \cdot10^{-5}
$$
Of course it still possible that there are other solutions outside just waiting for us to be discovered, a graph analysis might give some clearance about that. 
For the graph analysis please check the comment by Mark just below the answer! tl;dr: there are at most two solutions + there are at least two solutions $\to$ there are exactly two solutions 
