Exponential Decay — Definition, Formula & Examples

Q: What is the difference between exponential decay and exponential growth?

Both involve a quantity changing at a rate proportional to its current value, but in exponential growth the quantity increases over time (b > 1, or k > 0), while in exponential decay the quantity decreases (0 < b < 1, or k < 0). A growth curve rises steeply upward, whereas a decay curve falls toward zero without ever reaching it.

Q: How do you find the decay factor from a percentage?

If a quantity decreases by r percent each time period, the decay factor is b = 1 − r (where r is written as a decimal). For example, a 20% annual decrease gives b = 1 − 0.20 = 0.80. You then use A = A₀(0.80)ᵗ.

Q: Does an exponentially decaying quantity ever reach zero?

Mathematically, no. The function A = A₀bᵗ approaches zero as t increases but never equals zero. In practice, we treat the quantity as essentially gone once it becomes negligibly small. This is why scientists use half-life as a practical measure instead of waiting for the quantity to vanish entirely.

Exponential Decay

A model for decay of a quantity for which the rate of decay is directly proportional to the amount present. The equation for the model is A = A₀b^t (where 0 < b < 1 ) or A = A₀e^kt (where k is a negative number representing the rate of decay). In both formulas A₀ is the original amount present at time t = 0.

This model is used for phenomena such as radioactivity or depreciation. For example, A = 50e^–0.01t is a model for exponential decay of 50 grams of a radioactive element that decays at a rate of 1% per year.

Key Formula

A = A_0 \cdot b^{\,t} \quad (0 < b < 1) \qquad \text{or} \qquad A = A_0 \cdot e^{\,kt} \quad (k < 0)

Where:

$A$ = The amount remaining after time t
$A_0$ = The initial amount present at time t = 0
$b$ = The decay factor per time period, where 0 < b < 1
$t$ = Time elapsed
$k$ = The continuous decay rate (a negative number)
$e$ = Euler's number, approximately 2.71828

Worked Example

Problem: A car is purchased for $20,000 and depreciates at 15% per year. How much is the car worth after 6 years?

Step 1: Identify the initial value and the decay rate. The initial value is $20,000, and the car loses 15% of its value each year.

A_0 = 20{,}000, \quad \text{decay rate} = 15\%

Step 2: Find the decay factor b. Since 15% is lost each year, 85% remains, so b = 0.85.

b = 1 - 0.15 = 0.85

Step 3: Write the exponential decay equation.

A = 20{,}000 \cdot (0.85)^t

Step 4: Substitute t = 6 and compute.

A = 20{,}000 \cdot (0.85)^6 = 20{,}000 \cdot 0.37715 \approx 7{,}543

Answer: After 6 years the car is worth approximately $7,543.

Another Example

This example uses the continuous form A = A₀eᵏᵗ and shows how to derive the decay constant k from a given half-life, unlike the first example which used a directly given percentage rate.

Problem: A 200-gram sample of a radioactive substance has a half-life of 10 years. Using the continuous decay model A = A₀eᵏᵗ, find (a) the value of k and (b) how much remains after 25 years.

Step 1: Use the half-life condition: after 10 years, half the substance remains. Set up the equation with A = 100 and A₀ = 200.

100 = 200 \cdot e^{10k}

Step 2: Divide both sides by 200 and take the natural logarithm to solve for k.

e^{10k} = 0.5 \implies 10k = \ln(0.5) \implies k = \frac{\ln(0.5)}{10} \approx -0.0693

Step 3: Write the complete model with the computed k value.

A = 200 \, e^{-0.0693\,t}

Step 4: Substitute t = 25 to find the remaining amount.

A = 200 \, e^{-0.0693 \times 25} = 200 \, e^{-1.7325} \approx 200 \times 0.17678 \approx 35.36

Answer: After 25 years, approximately 35.4 grams of the substance remain.

Frequently Asked Questions

What is the difference between exponential decay and exponential growth?

Both involve a quantity changing at a rate proportional to its current value, but in exponential growth the quantity increases over time (b > 1, or k > 0), while in exponential decay the quantity decreases (0 < b < 1, or k < 0). A growth curve rises steeply upward, whereas a decay curve falls toward zero without ever reaching it.

How do you find the decay factor from a percentage?

If a quantity decreases by r percent each time period, the decay factor is b = 1 − r (where r is written as a decimal). For example, a 20% annual decrease gives b = 1 − 0.20 = 0.80. You then use A = A₀(0.80)ᵗ.

Does an exponentially decaying quantity ever reach zero?

Mathematically, no. The function A = A₀bᵗ approaches zero as t increases but never equals zero. In practice, we treat the quantity as essentially gone once it becomes negligibly small. This is why scientists use half-life as a practical measure instead of waiting for the quantity to vanish entirely.

Exponential Decay vs. Exponential Growth

	Exponential Decay	Exponential Growth
Definition	Quantity decreases at a rate proportional to its current value	Quantity increases at a rate proportional to its current value
Formula (base form)	A = A₀bᵗ where 0 < b < 1	A = A₀bᵗ where b > 1
Continuous form	A = A₀eᵏᵗ where k < 0	A = A₀eᵏᵗ where k > 0
Graph behavior	Curve falls, approaching zero asymptotically	Curve rises steeply without bound
Real-world examples	Radioactive decay, depreciation, cooling	Population growth, compound interest, viral spread

Why It Matters

Exponential decay appears across many subjects: in chemistry and physics (radioactive decay, Newton's law of cooling), in finance (asset depreciation), and in biology (drug elimination from the body). Algebra 2 and precalculus courses use it to introduce logarithms, since solving for time in a decay equation requires taking a logarithm. Understanding this model also prepares you for calculus, where exponential decay is a fundamental solution to differential equations.

Common Mistakes

Mistake: Using the decay rate directly as the base instead of subtracting it from 1. For instance, writing A = A₀(0.15)ᵗ when the decay rate is 15%.

Correction: The base b represents the fraction that remains, not the fraction lost. If 15% is lost each period, then 85% remains, so b = 1 − 0.15 = 0.85, and the model is A = A₀(0.85)ᵗ.

Mistake: Forgetting that k must be negative in the continuous model A = A₀eᵏᵗ, or dropping the negative sign during calculation.

Correction: For decay, k is always negative. If you compute k = ln(0.5)/10 ≈ −0.0693, keep the negative sign. A positive k would model growth, not decay, and give an increasing function.

Related Terms

Exponential Growth — The increasing counterpart to exponential decay
Half-Life — Time for a decaying quantity to halve
e — Base of the natural exponential used in continuous decay
Continuously Compounded Interest — Uses the same eᵏᵗ structure with k > 0
Logistic Growth — Growth model that slows and levels off, unlike pure exponential
Model — Exponential decay is a type of mathematical model
Direct Variation — Decay rate varies directly with amount present