Derivation of tau time-stepping in Gillespie algorithm?

I'm trying to find the derivation of tau ($\tau$) in the Gillespie algorithm. All the papers and chapters I've found simply say, without actually showing its derivation: "Tau is given by"

$\tau = \frac{1}{a_0(X_t)}ln\frac{1}{r_1}$

where $a_0$ is the propensity function, $X_t$ is the state vector and $r_1$ is one of two random numbers from the uniform distribution [0,1].

Specifically, I want to know where the $ln \frac{1}{r_1}$ comes from.

Does anybody know a good source where this is explained please?

The (or at least, one) derivation of tau is from interarrival times in a Poisson distribution. See, for example, chapter 5 (specifically, section 5.3.3 in the Ninth Edition) of Sheldon Ross's classic statistics text "Introduction to Probability Models."