Solve for $x$: $x =\ln(x)^4$

Question

I plotted the functions on both sides and it shows the equations has at least three solutions. Is there some non-interative (not sure if i used this term correctly - i mean the way you would solve, for instance, $x = x^2$) method to solve this equation?

Answer 1

Just for your information, this is a typical problem where Lambert function provides explicit solutions (which cannot be expressed in terms of elementary functions). This function $W(x)$ is defined according to $$x=W(x)e^{W(x)}$$

The equation you posted has five solutions but only three of them are real. These are given by $$x_1=e^{-4 W\left(\frac{1}{4}\right)}\approx 0.442394$$ $$x_2=e^{-4 W\left(-\frac{1}{4}\right)}\approx 4.17708$$ $$x_3=e^{-4 W_{-1}\left(-\frac{1}{4}\right)}\approx 5503.66$$

If you never heard about Lambert function, I suggest you have a look to Wikipedia page since it is a very interesting function (Euler and Lambert worked together) which is more and more used in many practical areas.

In practice, at least for the time being, just keep in mind that any equation which can write $$A+B x+C\log(D+Ex)=0$$ has explicit solutions in terms of this function.