Exponential change is change by the same percentage each step, rather than by the same amount. A population that grows 5% a year, a savings account earning interest, a phone battery losing charge, a medicine leaving the body: all exponential. Once you can read the formula, every one of these problems is the same problem.
Linear vs exponential
The quickest way to tell them apart is to look at how the output changes when the input goes up by 1.
| x | Linear: y = 100 + 20x | Exponential: y = 100(1.2)x |
|---|---|---|
| 0 | 100 | 100 |
| 1 | 120 | 120 |
| 2 | 140 | 144 |
| 3 | 160 | 172.8 |
| 10 | 300 | 619.2 |
- Linear adds the same amount each step (+20). The graph is a straight line, and its slope is the amount added. See the slope guide.
- Exponential multiplies by the same factor each step (×1.2). The graph curves, and it eventually outruns any line.
In a table, check differences for linear (constant difference) and ratios for exponential (constant ratio).
The formula
The general exponential model is:
y = a × bt
- a is the starting amount (the value when t = 0).
- b is the growth or decay factor, the number you multiply by each period.
- t is the number of periods.
If b > 1 the quantity grows. If 0 < b < 1 it decays.
When a problem gives a percentage rate r, the factor is:
- Growth by r: b = 1 + r. Growing 5% means b = 1.05.
- Decay by r: b = 1 − r. Shrinking 12% means b = 0.88.
5% growth is not b = 5 or b = 0.05
Write the rate as a decimal and add it to 1. A factor of 0.05 would mean losing 95% each period.
Example A town of 8,000 people grows 3% a year. Estimate the population after 10 years.
- a = 8,000 and b = 1 + 0.03 = 1.03.
- y = 8,000(1.03)10.
- 1.0310 ≈ 1.3439, so y ≈ 10,751 people.
Example A car bought for $28,000 loses 15% of its value each year. What is it worth after 4 years?
- a = 28,000 and b = 1 − 0.15 = 0.85.
- y = 28,000(0.85)4 ≈ 28,000 × 0.5220 ≈ $14,616.
Reading the formula backwards
Tests often give you the equation and ask what the numbers mean.
Example A model says P = 1,200(0.92)t, where t is years. What does each number mean?
- 1,200 is the starting value.
- 0.92 is less than 1, so it is decay. 1 − 0.92 = 0.08, so the value falls 8% per year.
If the exponent is not just t, the period changes. In y = 500(2)t/3, the amount doubles every 3 units of time, not every 1.
Compound interest
Interest that is added to the balance and then earns interest itself grows exponentially. When interest is compounded n times a year at annual rate r:
A = P(1 + r/n)nt
P is the starting principal and t is years.
Example $2,000 is invested at 6% annual interest, compounded monthly. How much is there after 5 years?
- P = 2,000, r = 0.06, n = 12, t = 5.
- A = 2,000(1 + 0.06/12)60 = 2,000(1.005)60.
- 1.00560 ≈ 1.3489, so A ≈ $2,697.70.
With continuous compounding, the formula uses the number e ≈ 2.71828:
A = Pert
The same $2,000 at 6% for 5 years, compounded continuously, gives 2,000e0.3 ≈ $2,699.72. More frequent compounding always helps, but less and less each time.
Half-life and doubling time
Radioactive decay and medicines are described by half-life, the time for half the amount to disappear. The formula is:
A = A₀(1/2)t/h
where h is the half-life.
Example A medicine has a half-life of 6 hours. A patient takes 400 mg. How much remains after 24 hours?
- t/h = 24/6 = 4 half-lives.
- 400 × (1/2)⁴ = 400/16 = 25 mg.
Doubling time works the same way with a factor of 2: A = A₀ × 2t/d.
A handy estimate is the rule of 72: at r% growth per period, something doubles in about 72/r periods. At 6% a year, money doubles in about 12 years.
Solving for time
When the unknown is in the exponent, you need logarithms. Take the log of both sides and bring the exponent down.
Example How long until $5,000 at 4% annual interest (compounded yearly) grows to $8,000?
- 8,000 = 5,000(1.04)t.
- Divide by 5,000: 1.6 = 1.04t.
- Take logs: t = log 1.6 / log 1.04 ≈ 0.2041/0.01703 ≈ 11.98.
- About 12 years.
The logarithms guide explains why this works. On the SAT, you can usually skip the logs by testing the answer choices or graphing in Desmos.
Problems to try
Is y = 3(0.7)x growth or decay, and by what percent?
Show answer
Decay by 30% per step. The factor 0.7 is less than 1, and 1 − 0.7 = 0.3.
A bacteria culture starts at 500 and doubles every hour. Write the formula and find the count after 6 hours.
Show answer
y = 500 × 2t. After 6 hours: 500 × 64 = 32,000.
A $900 laptop loses 20% of its value per year. What is it worth after 3 years?
Show answer
$460.80. 900(0.8)³ = 900 × 0.512.
$1,500 is invested at 5% compounded quarterly. Find the balance after 8 years.
Show answer
About $2,232.20. 1,500(1 + 0.05/4)32 = 1,500(1.0125)32 ≈ 1,500 × 1.4881.
Carbon-14 has a half-life of about 5,730 years. What fraction is left after 17,190 years?
Show answer
1/8. 17,190 ÷ 5,730 = 3 half-lives, and (1/2)³ = 1/8.
A table shows x = 0, 1, 2, 3 and y = 4, 12, 36, 108. Is it linear or exponential? Write the equation.
Show answer
Exponential, y = 4 × 3x. Each y is 3 times the one before.