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Exponential Growth

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- Exponential Growth Formula
- Calculating a Growing Population
- Solving for Growth Time

**Exponential Growth Formula**

The general exponential equation is y = a • b^{x}, where a base number, b, is raised to a variable exponent power, x. There is also a scalar multiplier, a, in front of the exponential expression.

Exponential growth can be modeled with a similar equation:

With exponential growth, b represents the rate of change (expressed as a decimal), and is added to one to indicate growth. We can call (1+b) the growth factor, since it's multiplied by a (the initial value) x number of times.

**Calculating a Growing Population **

According to the 2000 census, the population of the United States was 281.4 million people. The 2010 census reported a population of 308.7 million people, a growth of 9.7% over a period of 10 years. Assuming this rate of growth remains the same, what is the expected U.S. population in the year 2080?

From the scenario, we can draw the following values for certain variables in the exponential growth formula:

- a = 281.4 (million, the population in the year 2000)
- b = 0.097 (9.7% as a decimal)
- x = 8 (eight 10-year intervals between 2080 and 2000)

We could have alternatively used an a value of 308.7 (million) and an x value of 7.

It is helpful to include as many digits as possible during calculations, and then round at the end. This is especially helpful when your solution is a large number, such as a population in the hundreds of millions.

**Solving for Growth Time**

We can also use the exponential growth formula to calculate for a period of time. Suppose we want to find determine what year the United States is expected to reach a population of 400 million, given our earlier assumptions.

Many of our variables remain the same, but we no longer have a known value for x, as x represents time. Additionally, y now has a value of 400 (million). Solving for x in this case is going to require that we use logarithms, because it is the inverse operation of exponents:

In the last step, we multiplied x by 10, because x represents a 10-year time period between censuses.

There are a couple of things to note when solving these types of problems:

- Before doing anything about the variable exponent, divide the equation by the a-value. This is because the exponent is attached to the expression inside the parentheses (1+b), not a.
- You can apply the log of any base to undo the exponent, so long as the same base is used on both sides of the equation. More often than not, you will use either natural log (base e) or common log (base 10); either of which are fine.
- In applying the log to both sides, we use the power property of logarithms to bring the exponent outside of the log function, and place it as a scalar multiplier in front of the log. This allows us to isolate the variable through division.

Formulas to Know

- Exponential Growth