The function $g$ is defined by $g(x)=k\cdot a^x+b$, where $k,a,$ and $b$ are constants. In the $xy$-plane, the graph of $y=g(x)$ has a $y$-intercept at $(0,10)$ and passes through the points $(1,16)$ and $(2,28)$. What is the value of $a$?