Exponential Equations: Exponential Growth and Decay Application
A common application of exponential equations is to model exponential growth and decay such as in populations, radioactivity and drug concentration.
The formula for exponential growth and decay is:
EXPONENTIAL GROWTH AND DECAY FORMULA
y = ab^{x}
Where a ≠ 0, the base b ≠ 1 and x is any real number
In this function, a represents the starting value such as the starting population or the starting dosage level.
The variable b represents the growth or decay factor. If b > 1 the function represents exponential growth. If 0 < b < 1 the function represents exponential decay.
When given a percentage of growth or decay, determined the growth/decay factor by adding or subtracting the percent, as a decimal, from 1.
In general if r represents the growth or decay factor as a decimal then:
b = 1 – r Decay Factor
b = 1 + r Growth Factor
A decay of 20% is a decay factor of 1 – 0.20 = 0. 80
A growth of 13% is a growth factor of 1 + 0.13 = 1.13
The variable x represents the number of times the growth/decay factor is multiplied.
Let’s solve a few exponential growth and decay problems.
POPULATION
The population of Gilbert Corners at the beginning of 2001 was 12,546. If the population grew 15% each year, what was the population at the beginning of 2015?
Step 1: Identify the known variables.
Remember that the decay/growth rate must be in decimal form.
Since the population is said to be growing, the growth factor is b = 1 + r.

y = ? Population 2015
a = 12,546 Starting value
r = 0.15 Decimal form
b = 1 + 0.15 Growth Factor
x = 2015 – 2001 = 14 Years

Step 2: Substitute the known values.

y = ab^{x}
y = 12,546(1.15)^{14}

Step 3: Solve for y.

y = 88,772

RADIOACTIVITY
Example 1: The halflife of radioactive carbon 14 is 5730 years. How much of a 16 gram sample will remaining after 500 years?
Step 1: Identify the known variables.
Remember that the decay/growth rate must be in decimal form.
A halflife, the amount of time it takes to deplete half the original amount, infers decay. In this case b will be a decay factor. The decay factor is b = 1 – r.
In this situation x is the number of halflives. If one halflife is 5730 years then the number of halflives after 500 years is x=5005730x=5005730

y = ? Remaining grams
a = 16 Starting value
r = 50% = 0.5 Decimal form
b = 1 – 0.5 Decay Factor
x=5005730x=5005730 No. of Half lives

Step 2: Substitute the known values.

y = ab^{x}
y=16(0.5)5005730y=16(0.5)5005730

Step 3: Solve for y.

y = 15.1 grams

DRUG CONCENTRATION
Example 2: A patient is given a 300 mg dose of medicine that degrades by 25% every hour. What is the remaining drug concentration after a day?
Step 1: Identify the known variables.
Remember that the decay/growth rate must be in decimal form.
A drug degrading infers decay. In this case bwill be a decay factor. The decay factor is b = 1 – r.
In this situation xis the number of hours, since the drug degrades at 25% per hour. There are 24 hours in a day.

y = ? Remaining drug
a = 300 Starting value
r = 0.25 Decimal form
b = 1 – 0.25 Decay Factor
x = 24 Time

Step 2: Substitute the known values.

y = ab^{x}
y = 300(0.75)^{24}

Step 3: Solve for y.

0 = 0.30 mg
