Function $f$ is defined by $f(x)=c\cdot a^x+b$, where $a$, $b$, and $c$ are constants. In the $xy$-plane, the graph of $y=f(x)+10$ has a $y$-intercept at $(0,18)$. If $c=2$, and the product of $a$ and $b$ is $15$, what is the value of $a$?