# What is the name for an equation that has both an exponential variable and a variable in the base?

I would like to know if there is a name given to equations like the following:

$$-31=\frac{-39.2}{x}(1-e^{\frac{x}{4}})$$

or more generally:

$$a=\frac{b}{x}c^x$$

The formatting doesn't really matter, I'm just talking about an equation where there are variables both in an exponent and in another term in a base. Another example would be this:

$$20x^2 = e^x$$

I've searched for things like "composite exponential functions" but to no avail.

Also, I guess I could solve my example by rearranging so that there is an expression containing x on the left and an exponential one on the right. Then I could graph each side and see where they equal. However, is there a way to do it analytically?

Perhaps I'm not explaining this as clearly as I could be, but I would appreciate any advice you could offer.

Thanks!

Welcome to the world of Lambert function !

If you consider equation $$a=\frac{b \, e^{c x}}{x^d}$$ the solution is $$x=-\frac{d}{c} W\left(-\frac{c }{d}\left(\frac{b}{a}\right)^{1/d}\right)$$ where $$W(z)$$, the Lambert function, is such that $$z=W(z)\,e^{W(z)}$$.

The Wikipedia page will provide you many examples of the series of manipulations to be done.

In the real domain, $$W(z)$$ exists if $$z \geq -\frac 1 e$$ and, for $$z<0$$, it shows two branches.

Considering the example you give $$a=20$$, $$b=1$$, $$c=1$$, $$d=2$$, the result will be $$x=-2 W\left(-\frac{1}{4 \sqrt{5}}\right)$$ The Wikipedia page will also provide you series expansions for computing the value of $$W(z)$$. Using for example $$W(z)=z-z^2+\frac{3 z^3}{2}-\frac{8 z^4}{3}+\frac{125 z^5}{24}+O\left(z^6\right)$$ amking for your example $$x=\frac{7936+31321 \sqrt{5}}{307200}\approx 0.253815$$ while the exact value should be $$\approx 0.253871$$.

In fact, sooner or later, you will learn that any equation which can write $$A+Bx+C\log(D+Ex)=0$$ has solution(s) in terms of Lmabert function.