Sometimes "exponential growth" is just a figure of speech, a reference to anything that grows unreasonably or unbelievably quickly. But in certain cases, you can take the idea of exponential growth literally. For example, a population of rabbits can grow exponentially as each generation proliferates, then their offspring proliferate, and so on. Business or personal income can grow exponentially too. When you're called upon to make real-world calculations of exponential growth, you'll work with three pieces of information: Starting value, rate of growth (or decay), and time.

#### TL;DR (Too Long; Didn't Read)

**TL;DR (Too Long; Didn't Read)**

To calculate exponential growth, use the formula *y*(*t*) = *a__e ^{kt},* where

*a*is the value at the start,

*k*is the rate of growth or decay,

*t*is time and

*y*(

*t*) is the population's value at time

*t*.

## How to Calculate Exponential Growth Rates

Imagine that a scientist is studying the growth of a new species of bacteria. While he could input the values of starting quantity, rate of growth and time into a population growth calculator, he's decided to calculate the bacteria population's rate of growth manually.

If your rate of growth were to be less than 1, it tells you the population is shrinking. This is known as the rate of decay or the rate of exponential decay.

Looking back on his meticulous records, the scientist sees that his starting population was 50 bacteria. Five hours later, he measured 550 bacteria.

Inputting the scientist's information into the equation for exponential growth or decay, *y*(*t*) = *a__e*^{kt}, he has:

550 = 50_e^{k}_^{5}

The only unknown left in the equation is *k*, or the rate of exponential growth.

To begin solving for *k*, first divide both sides of the equation by 50. This gives you:

550/50 = (50_e^{k}_^{5})/50, which simplifies to:

11 = *e*^{_k_5}

Next, take the natural logarithm of both sides, which is notated as ln(*x*). This gives you:

ln(11) = ln(*e*^{_k_5})

The natural logarithm is the inverse function of *e ^{x}*, so it effectively "undoes" the

*e*function on the right side of the equation, leaving you with:

^{x}ln(11) = _k_5

Next, divide both sides by 5 to isolate the variable, which gives you:

*k* = ln(11)/5

You now know the rate of exponential growth for this population of bacteria: *k* = ln(11)/5. If you're going to do further calculations with this population – for example, plugging the rate of growth into the equation and estimating the population size at *t* = 10 hours – it's best to leave the answer in this form. But if you're not performing further calculations, you can input that value into an exponential function calculator – or your scientific calculator – to get an estimated value of 0.479579. Depending on the exact parameters of your experiment, you might round that to 0.48/hour for ease of calculation or notation.

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About the Author

Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! -- math subjects like algebra and calculus.

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