  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.
