Exponential Growth and Decay

Q: How do I convert a percent rate to the base b?

Write the percent as a decimal first (divide by $100$), then add it to $1$ for growth or subtract it from $1$ for decay. A $3\%$ growth rate gives $b = 1.03$; a $15\%$ decay rate gives $b = 0.85$. The most common mistake is skipping the decimal step and writing $1 + 8 = 9$ instead of $1 + 0.08 = 1.08$.

Linear change adds the same amount every step. Exponential change multiplies by the same amount every step — and that difference is enormous. A bacteria population that doubles every hour, a wildlife population shrinking $6\%$ per year, a radioactive sample losing half its mass every decade: all of these multiply by a fixed factor over and over, so they're modeled by exponential functions.

Every problem in this topic uses one formula: $y = a \cdot b^t$ , where $a$ is the starting amount, $b$ is the factor you multiply by each time period, and $t$ counts the time periods. If $b > 1$ the quantity grows; if $0 < b < 1$ it decays. Learn to read $a$ and $b$ out of a word problem and the rest is just plugging in.

Growth factor vs. rate

Word problems usually give you a rate $r$ as a percent, not the factor $b$ directly. The translation: growth means $b = 1 + r$ , and decay means $b = 1 - r$ , with $r$ written as a decimal. So the model becomes $y = a(1 + r)^t$ for growth and $y = a(1 - r)^t$ for decay.

Grows $8\%$ per year means $b = 1 + 0.08 = 1.08$ — the whole amount ( $100\%$ ) plus $8\%$ more. Shrinks $8\%$ per year means $b = 1 - 0.08 = 0.92$ — you keep $92\%$ of what you had. That second reading is the useful one: the base of a decay model tells you the fraction that remains each period.

This works in reverse too. Handed a model like $y = 1200(0.94)^t$ , read the base: $0.94 < 1$ , so it's decay, and since $1 - 0.94 = 0.06$ , the quantity loses $6\%$ per time period. The $1200$ is the starting amount, because plugging in $t = 0$ gives $y = 1200 \cdot 1 = 1200$ .

The table below shows a growth model that doubles each step, $y = 100(2)^t$ : notice each output is the one above times the factor $2$ , not the one above plus a fixed amount.

$t$	$y$
$0$	$100$
$1$	$200$
$2$	$400$
$3$	$800$

Spotting growth vs. decay instantly

You don't need a graph or a table — just look at the base. $y = 5(1.3)^t$ grows because $1.3 > 1$ . $y = 40(0.75)^t$ decays because the base is between $0$ and $1$ . A base of exactly $1$ would mean nothing changes at all.

One caution: exponential decay gets smaller and smaller but never reaches zero. Multiplying a positive amount by $0.9$ over and over shrinks it forever without ever hitting $0$ . On a graph, the curve flattens toward the horizontal axis without touching it.

Below, the solid curve $y = 1.5^x$ grows (base above $1$ ) while the dashed curve $y = 0.6^x$ decays (base between $0$ and $1$ ). Both pass through $(0, 1)$ , and both flatten toward the horizontal axis without ever crossing it.

Half-life

Half-life is decay with the cleanest possible factor: $b = \dfrac{1}{2}$ . If a substance has a half-life of $6$ hours, the amount is cut in half every $6$ hours. The model is $y = a\left(\dfrac{1}{2}\right)^n$ , where $n$ is the number of half-lives that have passed — not the number of hours. First convert the elapsed time into half-lives by dividing, then use the formula. Example 4 walks through it.

Worked examples

Example 1: bacteria growth

A dish starts with $400$ bacteria, and the population grows $25\%$ per hour. How many bacteria are there after $3$ hours?

Identify the starting amount and rate:

a = 400

r = 0.25

, growth

Build the model with

b = 1 + r

y = 400(1.25)^t

Substitute

t = 3

y = 400(1.25)^3

Compute the power:

1.25^3 = 1.953125

y = 400(1.953125)

Multiply

y = 781.25 \approx 781

Answer: About $781$ bacteria

Example 2: a shrinking sample

A $500$ -gram sample of a substance loses $10\%$ of its mass each day. How much remains after $4$ days?

Losing

10\%

means keeping

90\%

, so

b = 1 - 0.10 = 0.9

Build the model

y = 500(0.9)^t

Substitute

t = 4

y = 500(0.9)^4

Compute the power:

0.9^4 = 0.6561

y = 500(0.6561)

Multiply

y = 328.05 \approx 328

Answer: About $328$ grams

Example 3: reading a model

A population is modeled by $y = 1200(0.94)^t$ , where $t$ is in years. Is this growth or decay, what is the rate, and what is the starting population?

Check the base:

0.94

is between

0

and

1

, so this is decay

Find the rate from the base

r = 1 - 0.94 = 0.06

Convert to a percent: the population shrinks

6\%

per year

Find the starting amount by plugging in

t = 0

y = 1200(0.94)^0 = 1200

Answer: Decay at $6\%$ per year, starting from a population of $1200$ .

Example 4: radioactive half-life

A radioactive isotope has a half-life of $6$ hours. If a sample starts at $80$ mg, how much remains after $18$ hours?

Convert time to half-lives

n = \dfrac{18}{6} = 3

Build the model with

b = \dfrac{1}{2}

y = 80\left(\dfrac{1}{2}\right)^3

Compute the power

\left(\dfrac{1}{2}\right)^3 = \dfrac{1}{8}

Multiply

y = 80 \cdot \dfrac{1}{8} = 10

Sense check:

80 \to 40 \to 20 \to 10

across three half-lives ✓

Answer: $10$ mg

Try one yourself

Common questions

How do I convert a percent rate to the base b?

Write the percent as a decimal first (divide by $100$ ), then add it to $1$ for growth or subtract it from $1$ for decay. A $3\%$ growth rate gives $b = 1.03$ ; a $15\%$ decay rate gives $b = 0.85$ . The most common mistake is skipping the decimal step and writing $1 + 8 = 9$ instead of $1 + 0.08 = 1.08$ .

What's the difference between exponential and linear change?

Linear change adds a fixed amount each period: $200, 250, 300, 350$ . Exponential change multiplies by a fixed factor each period: $200, 300, 450, 675$ . To test data, check the differences between consecutive values (constant means linear) and the ratios (constant means exponential).

Can an exponential decay ever reach zero?

No. Each period keeps a fixed positive fraction of the amount, so the value gets closer and closer to zero without ever landing on it. That's why decay graphs flatten along the horizontal axis instead of crossing it.

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Exponential Growth and Decay

Growth factor vs. rate

Spotting growth vs. decay instantly

Half-life

Worked examples

Example 1: bacteria growth

Example 2: a shrinking sample

Example 3: reading a model

Example 4: radioactive half-life

Try one yourself

Common questions

Keep learning

Product & Quotient Rules

Solving Exponential Equations & Inequalities

Introduction to Functions