Function $f$ is defined by $f(x)=A\cdot B^x+C$, where $A$, $B$, and $C$ are constants. In the $xy$-plane, the graph of $y=f(x)$ has a y-intercept at $(0,15)$. The graph of $y=f(x)-7$ has a y-intercept at $(0,8)$. If $A=3$, what is the value of $C$?