I'm a little rusty with my math and have forgotten a lot of techniques for solving equations. I tried using $\ln$ to get rid of $e^x$ but then I end up with $\ln(x^3+\frac{13}{4})=x$ and I'm stuck.
I was able to approximate the solutions by using a graphing calculator to see that there were in fact two solutions and then applying Newton's method. However, I was wondering if there was an analytical solution I'm just not seeing.
As a side question (if there is no analytical solution), is there a quick way to determine the number of solutions without resorting to a calculator?
If $x$ is algebraic (that is, if $x$ is a solution of a polynomial equation with integer coefficients) and not zero, then there's a theorem that says $e^x$ is not algebraic, so $x$ can't be a solution of your equation. By the same token, if $e^x$ is algebraic and not $1$ then $x$ is not algebraic, so $x$ can't be a solution of your equation. So we've pretty much ruled out as possible solutions the numbers you're most familiar with like $\sqrt2$ and $\log 2$. There are more advanced techniques to show that your equation can't have a solution in closed form in terms of the familiar functions of school mathematics; only numerical methods are available.