Is it possible to solve exponential equation analytically? I'm trying to solve the following equation:
$$e^{3x}-e^{2x}\left(e^2-\frac{1}{e^4}\right)-1=0$$
I know the solution is 2, as the equation above is simply a rearranged version of this initial statement:
$$e^{x}-\frac{1}{e^{2x}}=e^2-\frac{1}{e^4}$$
I assumed I could forge a cubic by letting $x=e^b$ and then using the cubic formula to do so, but I get into a hideous mess with terms being "trapped" inside cube roots and nothing really falls together nicely.
My question is, how would one go about solving this equation analytically (if it's at all possible)?

With $t:=e^x$ and mutliplying by $e^4$,
$$e^4t^3-t^2\left(e^6-1\right)-e^4=0$$
indeed has the solution $t=e^2$. By long division,
$$e^4t^3-t^2\left(e^6-1\right)-e^4=(t-e^2)\left(e^4t^2+ t+e^2\right)=0.$$
Now you can solve the quadratic.
