The growth rate in an exponential function is the percentage by which a quantity increases during each time period. In the standard formula f(x) = a(1 + r)x, the growth rate is r, expressed as a decimal. A growth rate of 0.10, for example, means the quantity grows by 10% each period.
How Growth Rate Fits Into the Formula
Exponential functions come in two common forms, and the growth rate plays a slightly different role in each.
The first is the discrete form: f(x) = a(1 + r)x. Here, a is the starting value, r is the growth rate as a decimal, and x is the number of time periods. If you start with 500 and grow at 8% per year, the function is f(x) = 500(1.08)x. After one year the value is 540; after two years it’s 583.20. Each year, the amount added gets larger because you’re always taking 8% of the new, bigger total.
The second is the continuous form: f(t) = a · ert. This version uses the mathematical constant e (roughly 2.71828) and assumes growth happens constantly rather than in periodic jumps. The variable r still represents the growth rate, and t is time. Scientists and economists use this form when modeling things like population growth or radioactive decay, where change doesn’t wait for the end of a neat time interval.
Growth Rate vs. Growth Factor
These two terms sound similar but mean different things, and mixing them up is a common source of confusion. The growth rate is the percentage increase per period, while the growth factor is the number you multiply by each period. They’re connected by a simple relationship: the growth factor equals 1 + r.
If an investment yields 10% annually, the growth rate is 0.10 (or 10%) and the growth factor is 1.10. When you see an exponential function written as f(x) = a · bx, the value b is the growth factor. To extract the growth rate from it, just subtract 1. A growth factor of 1.25 means a growth rate of 25%. A growth factor of 0.85 means the quantity is shrinking by 15% per period, which brings us to decay.
When the Growth Rate Is Negative
A negative growth rate produces exponential decay instead of exponential growth. In the continuous form f(t) = a · ekt, a positive k means the quantity grows over time and a negative k means it shrinks. In the discrete form, a negative r makes the growth factor (1 + r) less than 1, so the value gets smaller with each period. A radioactive substance losing 5% of its mass per year has a growth rate of r = −0.05 and a growth factor of 0.95.
How to Calculate the Growth Rate
If you know a starting value and an ending value over a certain time span, you can solve for r directly. Using the continuous model, the formula rearranges to:
r = ln(N(t) / N0) / t
Here, N0 is the starting amount, N(t) is the amount after time t, and ln is the natural logarithm. Say a population grows from 2,000 to 5,000 over 6 years. Dividing 5,000 by 2,000 gives 2.5; the natural log of 2.5 is about 0.916; dividing by 6 gives a continuous growth rate of roughly 0.153, or 15.3% per year.
If you have two data points on a graph and want to find the growth factor for a discrete model (y = a · bx), plug both points into the equation to create two separate equations, then divide one by the other. This cancels out the starting value a, leaving you with b. For instance, if the points (2, 4) and (4, 10) both satisfy the equation, dividing gives 10/4 = b4/b2, so b2 = 2.5 and b ≈ 1.58. The growth rate is then 1.58 − 1 = 0.58, or 58% per period.
Growth Rate in Compound Interest
Compound interest is one of the most practical applications of exponential growth rates. The formula A(t) = P(1 + r)t describes an account where interest compounds once a year: P is your initial deposit, r is the annual interest rate, and t is the number of years.
When interest compounds more frequently, the formula adjusts to A(t) = P(1 + r/n)nt, where n is the number of compounding periods per year. This matters because compounding more often produces a slightly larger result even with the same stated rate. An account advertising 12% compounded monthly doesn’t just apply 12% once at year’s end. It applies 1% (that’s 12% divided by 12) each month, and each month’s interest earns interest in the following months. The effective annual growth factor becomes (1 + 0.12/12)12 ≈ 1.1268, so the effective annual growth rate is about 12.68%, not 12%. The difference between the stated (nominal) rate and the effective rate grows larger as compounding becomes more frequent.
Doubling Time and the Rule of 70
One of the most useful shortcuts for working with exponential growth rates is the Rule of 70. To estimate how long it takes a quantity to double, divide 70 by the growth rate expressed as a percentage. An investment growing at 6% per year doubles in roughly 70 / 6 = 11.7 years. A population growing at 2% doubles in about 35 years.
This works because of the logarithmic relationship buried inside the doubling formula. The exact doubling time is ln(2)/r, and ln(2) is approximately 0.693, which is close enough to 0.70 that dividing 70 by the percentage rate gives a reliable estimate. It’s not precise for very high growth rates, but for single-digit percentages it’s remarkably accurate.
Exponential Growth Rates in Nature
Biology offers vivid examples of how different growth rates play out. The bacterium E. coli can divide every 20 minutes under ideal lab conditions, which corresponds to a per-hour growth rate of roughly 200%. In natural environments, though, the same species doubles only about every 15 hours because nutrients are scarcer and conditions are harsher. That’s a roughly 45-fold difference in doubling time between the lab and the wild.
Other species span an enormous range. Vibrio cholerae doubles roughly every 1.1 hours in the wild, making it one of the faster-growing bacteria outside a lab. Staphylococcus aureus doubles approximately every 1.9 hours. At the slow end, the bacterium that causes leprosy doubles every 300 to 600 hours (about 12 to 25 days) on mouse tissue. These differences all come down to the growth rate r in the exponential model. A higher r means faster doubling, and the relationship is not linear: even small changes in r dramatically alter how quickly a population explodes.
This is the defining feature of exponential growth. Unlike linear growth, where the same fixed amount is added each period, exponential growth adds a fixed percentage of the current total. Early on, the increases look modest. But as the base grows, each percentage increase represents a larger and larger absolute jump, which is why exponential curves appear to “take off” suddenly after a period of seemingly slow change.