What is x in terms of p?

Question

In this equation

$$\frac x{\ln(x)} = p$$

How can I have $x$ as a function of $p$?

Answer 1

This is a typical example of a transcendental equation with a solution that cannot be expressed with most widely known analytical functions (trigonometric, exponential and logarithmic).

Rule of thumb: if $x$ is found both inside and outside log/exp/sin, the equation cannot be inverted in terms of these functions.

However, the equation is common enough to have a so called "special function" which can be used to invert it: Lambert's function $x=W(y)$, which solves equation $xe^{x}=y$.

Of course, as Lambert's function doesn't usually have a button on a calculator, that does not help much, but does mean people have studied this function and have efficient means of calculating its value.

You just have to transform it into the form to apply the rule for Lambert's function;

$$\ln x = \frac{x}{p}$$ $$x=e^{x/p}$$ $$xe^{-x/p}=1$$ new variable $u=-x/p$ $$-pu e^{u}=1$$ $$ue^u=-1/p$$ $$u=W(-1/p)$$ $$x=-pu=-p W(-1/p)$$

Of course you have to be careful because for negative arguments, $W$ can have two branches (equation has two solutions), or no real solution at all, if the value is too negative (below $-1/e$).