NO. Please stop perpetuating this. It's wrong.

Just because an equation has an exponent in it doesn't mean that it's an exponential function.

Power is a cubic function of speed, meaning P=a*v^3 + b*v^2 + c*v + d. Finding a, b, c, and d is sometimes difficult, but at high speeds, the v^3 term is much more important than the other terms, so they often get ignored, and rightly so.
Moral: going twice as fast requires 8 times as much power.

Force is a quadratic (square) function of speed, so F = a*v^2 plus the lower order terms.
Moral: Going twice as fast requires 4 times as much force to the pedals, but of course you have to turn those pedals twice as fast, hence the 8x from the power equation.

An exponential function is one where the variable in question IS the exponent. Completely unrelated to cycling, but bank interest, bacteria growth, and radioactive decay are all exponential functions with the form Y=a^x where (a) is a constant and (x) is a variable.