I want to solve the following exponential equation:
$$\frac{e^x}{x}=\frac{2e^{\frac 3 2}}{3}$$
Now, I think it is easy enough to see in the head that the answer is $x=\frac 3 2$. Though, I wanted to try to solve it in the standard way. I multiply both sides by $x$:
$$e^x=\frac{2xe^{\frac 3 2}}{3}$$
Now I apply the logarithm function to both sides:
$$\ln(e^x)=\ln\left(\frac{2xe^{\frac 3 2}}{3}\right)$$
Which should be equal to
$$x=\ln\left({2xe^{3/2}}\right)-\ln{3}\rightarrow \dots$$
I can't really seem to get to the solution. Any hints on what I'm doing wrong?
Best Answer
Transcendental equations usually have no closed-form solution, except in coincidental cases (like yours), and there is no systematic way to solve. So you are doing nothing wrong.
When you can transform the equation in a polynomial one, then you can see if the equation is solvable analytically, as the case of polynomials has been settled. Otherwise, you are on your own.
Anyway, for your particular equation, there is a special function (Lambert's $W$) that you can use: it solves $xe^x=y$ for $x$ and its properties have been studied. Your equation can be turned to that form. (At the same time this "proves" that your equation cannot be solved by other means.)
Also note that numerical solvers can find approximations of the roots and may allow you to recognize the exact value if there is one. But I wouldn't recommend this as a routine approach.