At most one solution to $a^x + b^y = c^z$ for some ranges of $a$, $b$, $c$

We consider the number of solutions in positive integers $(x,y,z)$ for the purely exponential Diophantine equation $a^x+b^y =c^z$ (with $\gcd(a,b)=1$). Apart from a list of known exceptions, a conjecture published in 2016 claims that this equation has at most one solution in positive integers $x$, $y$, and $z$. We show that this is true for some ranges of $a$, $b$, $c$, for instance, when $1 < a,b < 3600$ and $c<10^{10}$. The conjecture also holds for small pairs $(a,b)$ independent of $c$, where $2 \le a,b \le 10$ with $\gcd(a,b)=1$. We show that the Pillai equation $a^x - b^y = r > 0$ has at most one solution (with a known list of exceptions) when $2 \le a,b \le 3600$. Finally, the primitive case of the Je\'smanowicz conjecture holds when $a \le 10^6$ or when $b \le 10^6$. This work highlights the power of some ideas of Miyazaki and Pink and the usefulness of a theorem by Scott.