There are many situation that may be modeled by an exponential function. Hence, the population growth or exponential decay are some of the problems that may be represented as exponential functions.
As an example, I suggest to you to look at the equation used to model the population growth or exponential decay: `P = P_0*e^(kt)`
`P_0` expresses the population at the beginning
P expresses the population at the moment of computation
k expresses the growth or decay
t expresses the amount of time between the start and the end moments.
Other common application of exponential function is computation of different kinds of investments.
As an example, you should examine the structure of the formula that expresses the compound interest such that:
`A = P(1 + r/n)^(nt)`
P expresses the initial amount of money
A expresses the amount of money aftera time t
r expresses the interest rate and n tells how many times the interest is compounded along one year.
t expresses the time measured in years
Hence, you need to remember that exponential function is very effective when it is used to model various real situations; exponential decay, population growth or compound interest being only some of these practical applications.
Exponenial functions have many more real-world uses than a lot of mathematical functions do. Increasing exponential functions (for an example, type 2^x into your graphing calculator; these functions increase more and more sharply over time) can model population growth very well. Think about population of a city: if the population is 1000 in one year, it's not likely to be 1020 the next, 1040 the next, 1060 the next, and so on. That would be a linear function (just a line), and population tends not to work that way! It's much more likely to grow slowly at first and then really take off.
As a smaller (and easier to understand) example, take bacteria. If we start off with just 2 bacteria that multiply every hour, in one hour we'll have 4. In another hour, thouh, we won't have 6--we'll have 8! It doubled again. In another hour, we'll have 16. In this way, the population grows very fast, and an exponential function best models that growth.
Exponential decay (when the exponent is negative instead of positive) probably best models radioactive decay. If an element has a half life of 10 years, that means that a 100g sample of the element will be at 50g after ten years. After another ten years, it's down to 25g, because 25 is half of 50. After another ten years (so it's been 30 years now), we have 12.5g. As you can see, the amount that the sample decreases gets less and less over time, just like an exponential function.