Intersection of an exponential function and a line

An intersection of an exponential function and a horizontal line amounts to solving an exponential equation. We find where $y = 2^x$ meets the line $y = 4$.

At an intersection $2^x = 4$. Writing the right side as a power of the same base, $4 = 2^2$, gives $2^x = 2^2$. Once the bases match, the exponents must be equal, so $x = 2$. The intersection is the single point $(2, 4)$.

The exponential $y = 2^x$ has a base greater than $1$, so it is always strictly increasing. Hence it meets a horizontal line $y = k$ at exactly one point for any $k > 0$. Since its range is $y > 0$, it does not meet a line with $k \leq 0$.

So an increasing graph meets a horizontal line in at most one point, and the exponential equation $2^x = k$ (with $k > 0$) always has exactly one solution. The large dot on the graph is the intersection $(2, 4)$.