If something grows at a rate of 25% per month, then the amount after 16 months will be $ P\cdot1.25^{16}$. This is true of money, of population, or of anything else. The phrase "grows at a rate of 25% per month" means that at the end of every month the quantity is 25% bigger than it was at the beginning of the month. The factor of increase for one month is therefore 1.25, and the factor of increase for 16 months is $1.25^{16}$.
It is always possible to rewrite the expression for quantity in a base other than base 1.25. The usual choice is the natural base $e\approx2.718$. In this base, the amount is given by $P\cdot e^{16k}$, where $k=\log_e1.25\approx0.2231$. But $k$ is not the same thing as the rate. Sometimes $k$ is referred to as the "rate under continuous compounding" or "continuous rate". This terminology comes from the theory of compound interest. So, for example, if you invest \$15000 at an annual rate of 20%, compounded continuously, then the amount after three years will be $15000\cdot e^{(3)(0.2)}$, whereas without the continuous compounding it will be $15000\cdot1.2^3$. If you are not familiar with compound interest, you can simply think of $k$ as the base conversion factor - that is, the factor needed to convert from base 1.25 to base $e$.
You always need to be clear about what kind of rate you are dealing with. If the word "rate" is used without a modifier such as "continuous", you should assume that it is the ordinary sort of rate described in the first paragraph above.