By that definition you can turn any linear function a * x + b, "exponential" by making it e^ln(a*x +b) even though it's actually linear (you can do it to anything, including sin() or even ln() itself, which would make per that definition the inverse of exponential "exponential").

Essentially you're just doing f(f^-1^(g(x))) and then saying "f(m) is e^m^ so if I make m = ln(g(x)) then g(x) is exponential"

Also the correct formula in your example would be e^(ln(2)*X/3) since the original formula if X denotes months is 2^X/3^