Great question!!
Compounded "continuously" is only for money and the question must specify that the interest the bank gives is compounded "continuously. That is the only context in which we use the P(t) = P0ert formula.
When an annual growth (or depreciation) rate is given as a %, that is when we use (1 + r) (or (1 - r)) as the base of the exponential function: A(t) = A0(1 + r)t. A variant of this formula is used for compounding interest n times per year. If we see wording like "compounded monthly" or "semiannually," then we use the following:
A(t) = A0(1 + r/n)nt.
Other word problems call for other formulas. Exponential decay for which a 1/2-life is given is modeled with
A(t) = A0(1/2)t/k , where k is the length of the half-life of the substance in years.
If we know an amount doubles every k years, then A(t) = A0(k√2)t is useful, which can also be written as
A(t) = A0(2)t/k.
Finally, in calculus, we frequently like to employ e, Euler's constant, as the base of an exponential growth function, as this is the only base that generates a function whose growth rate equals its y-value.
Regardless of which of the above we use, exponential growth and decay is characterized by situations in which the rate of change (growth or decay) is proportional to the amount there (the y-value). Thus, exponential growth continues to grow faster and faster. Exponential decay decreases more slowly all the time. There is also a very close connection between exponential functions and geometric sequences.
