Exponential Growth

Learn how quantities grow exponentially and apply growth formulas to real-world scenarios.

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Definition

Exponential growth occurs when a quantity increases by a constant percentage (or factor) over equal time intervals. The general formula is:
A(t)=A0⋅btA(t) = A_0 \cdot b^t
Where:
  • A(t)A(t) = amount after time tt
  • A0A_0 = initial amount (at t=0t = 0)
  • bb = growth factor (base), where b>1b > 1
  • tt = time
Alternatively, with a percentage growth rate rr:
A(t)=A0⋅(1+r)tA(t) = A_0 \cdot (1 + r)^t
The key characteristic of exponential growth is that the rate of increase accelerates over time. Unlike linear growth (adding a constant), exponential growth multiplies by a constant.

Try it now

Which formula represents exponential growth?

Worked Examples

A bacteria colony starts with 500 cells and doubles every hour. How many cells will there be after 6 hours?
1

Identify the values

A0=500A_0 = 500 (initial), b=2b = 2 (doubles), t=6t = 6 hours → Values identified
2

Write the formula

A(t)=A0⋅bt=500⋅26A(t) = A_0 \cdot b^t = 500 \cdot 2^6 → Formula set up
3

Calculate 262^6

26=2×2×2×2×2×2=642^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 → 26=642^6 = 64
4

Multiply

A(6)=500⋅64=32000A(6) = 500 \cdot 64 = 32000 → 32000 cells

Common Mistakes

Confusing growth factor with growth rate

Why it's wrong: If something grows by 5%, the growth rate is r=0.05r = 0.05, but the growth factor is b=1.05b = 1.05. Students often use 0.05 as the base instead of 1.05.

Correct: Growth factor = 1+1 + growth rate. If the rate is 5%, use b=1.05b = 1.05 in the formula.

Thinking exponential growth is always fast

Why it's wrong: Early exponential growth can look slow. 21=22^1 = 2, 22=42^2 = 4, 23=82^3 = 8 seems modest, but it accelerates dramatically.

Correct: Exponential growth starts slowly but accelerates. The magic is in the later stages: 220>1,000,0002^{20} > 1,000,000!

Using addition instead of multiplication

Why it's wrong: Linear thinking is intuitive: 'Add 10% each year' but exponential growth multiplies: 100→110→121→133.1100 \to 110 \to 121 \to 133.1

Correct: Each period, multiply by the growth factor: Anew=Aold⋅bA_{new} = A_{old} \cdot b

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Interactive Visual

Function graph

012345670100002000030000xy

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Table of values

x0123456
a·b^x50010002000400080001600032000
linear500100015002000250030003500

Sequence Explorer

1,
2,
4,
8,
16,
32,
64,
128
, ...

Formula

a_n = 1 × 2^(n-1)
12345678
a_1

1

a_5

16

a_10

512

a_20

524,288

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Interactive Grapher

-10-50510-10-50510xy

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y = 2^x

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Practice Problems

16 problems
Problem 1 of 16
Easy
Which formula represents exponential growth?

Why It Matters

Exponential growth appears in many critical real-world situations:
  • Population Biology: Bacteria can double every 20 minutes under ideal conditions
  • Finance: Compound interest grows your savings exponentially over decades
  • Technology: Computing power has doubled roughly every 2 years (Moore's Law)
  • Epidemiology: Disease spread in early outbreak phases follows exponential patterns
  • Social Media: Viral content spreads exponentially through networks
Understanding exponential growth helps you make informed decisions about investments, recognize unsustainable growth patterns, and appreciate how quickly things can change.

Real World Applications

Compound Interest and Retirement

Banks use exponential growth for compound interest. Starting early makes a huge difference due to the 'snowball effect'.

Example:

Investing 5000 euros at 7% annual return: after 30 years you'll have 5000⋅1.0730≈38,0615000 \cdot 1.07^{30} \approx 38,061 euros!

1Try It Yourself

You invest 2000 euros at 6% annual interest.

How much will you have after 15 years?

Step 1: Write the mathematical expression

Use the formula: 2000⋅(1.06)152000 \cdot (1.06)^{15}

Population Growth Models

Biologists model population growth under ideal conditions (unlimited resources) as exponential.

Example:

If a rabbit population of 100 grows at 50% per year: after 4 years there would be 100⋅1.54=506100 \cdot 1.5^4 = 506 rabbits.

2Try It Yourself

A fish population of 800 grows at 25% per year.

What will the population be after 3 years?

Step 1: Write the mathematical expression

Calculate: 800⋅(1.25)3800 \cdot (1.25)^3

Technology and Moore's Law

Gordon Moore observed that computing power doubles roughly every 2 years, an exponential pattern that held for decades.

Example:

If a computer has 1 million transistors today and doubles every 2 years, in 20 years it will have 1,000,000⋅210≈11,000,000 \cdot 2^{10} \approx 1 billion transistors.

3Try It Yourself

A smartphone has 10 billion transistors. Assume doubling every 2 years.

How many transistors in 10 years?

Step 1: Write the mathematical expression

Calculate: 10⋅2510 \cdot 2^5 billion

Key Takeaways

  • 1Exponential growth follows the formula A(t)=A0⋅btA(t) = A_0 \cdot b^t, where b>1b > 1
  • 2The growth factor b=1+rb = 1 + r, where rr is the growth rate (as a decimal)
  • 3Exponential growth accelerates over time unlike linear growth
  • 4Doubling time is the period for a quantity to double: if b=2b = 2, one period doubles the amount
  • 5Real applications include compound interest, population biology, and technology advancement

Frequently Asked Questions

Linear growth adds a constant amount each period (like +10+10 per year). Exponential growth multiplies by a constant factor (like ×1.1\times 1.1 per year). Over time, exponential growth far outpaces linear growth.
Linear growth adds a constant amount each period (like +10+10 per year). Exponential growth multiplies by a constant factor (like ×1.1\times 1.1 per year). Over time, exponential growth far outpaces linear growth.
Doubling time is how long it takes for a quantity to double. For 7% annual growth, the doubling time is approximately 707=10\frac{70}{7} = 10 years (Rule of 70).
In the real world, no. Resources are limited, so exponential growth eventually slows (called logistic growth). But the exponential model works well for early growth phases.

Glossary

Exponential growth
Growth where a quantity increases by a constant percentage over equal time periods
Growth factor
The multiplier bb in the formula; equals 1+1 + growth rate
Growth rate
The percentage increase per time period, expressed as a decimal (rr)
Doubling time
The time required for a quantity to double in value
Initial value
The starting amount A0A_0 at time t=0t = 0

Formula Card

Exponential Growth Formula

A(t)=A0⋅(1+r)tA(t) = A_0 \cdot (1 + r)^t

$A(t)$ = amount after time $t$, $A_0$ = initial amount, $r$ = growth rate (as decimal), $t$ = time periods. When $r > 0$, the quantity grows.

Today's challengeOne problem, three tries

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