Number of solutions to a special type of unit equations in two unknowns, III

It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. We develop the methods in our previous work which rely on a variety from Baker's theory and thoroughly study the conjecture for cases where $c$ is small relative to $a$ or $b$. Using restrictions derived under which there is more than one solution to the equation, we obtain a number of finiteness results on the conjecture, which in particular enables us to find some new values of $c$ being presumably infinitely many such that for each such $c$ the conjecture holds true except for only finitely many pairs of $a$ and $b$. Most importantly we prove that if $c=13$ then the equation has at most one solution, except for $(a,b)=(3,10)$ or $(10,3)$ which exactly gives two solutions. Further our study with the help of Schmidt Subspace Theorem among others brings strong contributions to the study of Pillai's type Diophantine equations, which includes a general and satisfactory result on a well-known conjecture of M. Bennett on the equation $a^x-b^y=c$ for any fixed positive integers $a,b$ and $c$ with both $a$ and $b$ greater than 1. Some conditional results are presented under the $abc$-conjecture as well.