A Recurring Problem

A recurring problem I keep running into is solving

$$ax+b=x^c$$

As far as I can tell, there is no general closed-form solution for $x$, so I'm maintaining this page as a compilation of things I know about it.

Closed-Form Solutions

The closed-form solutions for $a$ and $b$ are obvious. $x$ has a few closed-form solutions when $c$ is a particular fixed value:

$c$solution
$$c=-0.5$$some wonky cubic solutions
$$c=0$$$$ x = \frac{1-b}{a} $$
$$c=0.5$$$$ \frac{1-2ab \pm \sqrt{1 - 4ab}}{2a^2} $$
$$c=1$$$$ x = \frac{b}{1-a} $$
$$c=2$$$$ \frac{a \pm \sqrt{a^2+4b}}{2} $$

Responsiveness

We know that

$$ x^c-ax-b=0 $$

so

$$ \left( c x^{c-1} - a \right) \delta x - \left( x \right) \delta a - \delta b = 0 $$

so

$$ \delta x = \frac{ x \cdot \delta a + \delta b}{c x^{c-1} - a} $$

In particular, at $x = 0$, we have

$$ \delta x = \frac{\delta b}{a} $$

and at $x = 1$, we have

$$ \frac{\delta a + \delta b}{c - a} $$

Other Properties

  • If $x=0$, then $b = 0$
  • If $x=1$, then $a + b = 1$