4.7Exponential and Logarithmic Models
Apply exponential growth and decay, continuous compound interest, logistic growth, and Newton's Law of Cooling to real-world problems. Identify the right model, find the growth constant, and answer questions about the quantity over time.
Every field that tracks change over time — biology, finance, medicine, physics, environmental science — relies on exponential and logarithmic models. Understanding these models is the bridge between pure math and real-world problem solving.
Essential Question
How do exponential and logarithmic models translate real-world phenomena — population growth, radioactive decay, compound interest, and disease spread — into equations we can analyze and solve?
Lesson Overview
Real-world quantities that grow or decay at a rate proportional to their current size are modeled by exponential functions. When growth is unlimited, we use A(t) = A₀eᵏᵗ. When growth is constrained by a maximum (carrying capacity), we use the logistic model. Finance uses the continuous compound interest formula A = Pe^(rt). Cooling and heating follow Newton's Law of Cooling. Quantities that grow slowly after an initial surge are often modeled with a logarithmic function. In every case, the key skill is identifying which model applies, setting up the equation, finding the growth/decay constant k, and then answering questions about the quantity at a given time.
Five model types — identify which applies before setting up any equation
Finding k — The Key Step
- You need two data points to find k
- Plug both into A(t) = A₀eᵏᵗ
- Divide the two equations to eliminate A₀
- Solve for k using ln
- k > 0 → growth; k < 0 → decay
Model Selection Guide
- Unlimited growth: A(t) = A₀eᵏᵗ
- Radioactive / drug decay: A(t) = A₀eᵏᵗ, k < 0
- Money / continuous interest: A = Pe^(rt)
- Population with limit: logistic model
- Slow growth after surge: f(t) = a + b·ln(t)
- Cooling / heating: Newton's Law
Worked Examples
A bacterial culture starts with 500 cells and doubles every 3 hours. Write an exponential growth model and find the population after 10 hours.
Model: A(t) = A₀·eᵏᵗ with A₀ = 500
Find k using the doubling condition: A(3) = 1000
500·e^(3k) = 1000 → e^(3k) = 2 → 3k = ln(2) → k = ln(2)/3 ≈ 0.2310
Model: A(t) = 500·e^(0.2310t)
At t = 10: A(10) = 500·e^(0.2310·10) = 500·e^2.310 ≈ 500·10.077 ≈ 5039 cells
Carbon-14 has a half-life of 5730 years. A fossil contains 30% of its original carbon-14. How old is the fossil?
Model: A(t) = A₀·eᵏᵗ. Use half-life to find k.
A(5730) = A₀/2 → e^(5730k) = 1/2 → 5730k = ln(1/2) = −ln(2)
k = −ln(2)/5730 ≈ −0.0001210
Set A(t) = 0.30·A₀: 0.30·A₀ = A₀·e^(kt) → 0.30 = e^(kt)
ln(0.30) = kt → t = ln(0.30)/k = −1.2040/(−0.0001210) ≈ 9950 years
$8000 is invested at 4.5% annual interest compounded continuously. How long until the investment doubles?
Model: A = Pe^(rt) with P = 8000, r = 0.045
Double: A = 16000 → 16000 = 8000·e^(0.045t)
e^(0.045t) = 2 → 0.045t = ln(2)
t = ln(2)/0.045 = 0.6931/0.045 ≈ 15.4 years
Rule of 70 check: 70/4.5 ≈ 15.6 years ✓ (close approximation)
A cup of coffee at 90°C is placed in a 22°C room. After 5 minutes it cools to 70°C. Find the temperature after 20 minutes.
Newton's Law of Cooling: T(t) = T_s + (T₀ − T_s)·eᵏᵗ
T_s = 22, T₀ = 90: T(t) = 22 + 68·eᵏᵗ
Use T(5) = 70: 70 = 22 + 68·e^(5k) → 48 = 68·e^(5k) → e^(5k) = 48/68
5k = ln(48/68) → k = ln(48/68)/5 ≈ −0.0693
T(20) = 22 + 68·e^(−0.0693·20) = 22 + 68·e^(−1.386) = 22 + 68·0.25 = 22 + 17 = 39°C
A logistic model for a town's population is P(t) = 50000/(1 + 24·e^(−0.15t)), where t is years since 2000. Find the population in 2020 and the carrying capacity.
Carrying capacity c = 50000 (numerator of logistic model)
t = 2020 − 2000 = 20
P(20) = 50000/(1 + 24·e^(−0.15·20)) = 50000/(1 + 24·e^(−3))
e^(−3) ≈ 0.0498 → 24·0.0498 ≈ 1.195
P(20) = 50000/(1 + 1.195) = 50000/2.195 ≈ 22779
Guided Practice
A population of 2000 rabbits grows to 3500 in 4 years. Write an exponential growth model and predict the population after 10 years.
Hint: Use A(4) = 3500 with A₀ = 2000 to find k. Then evaluate A(10).
A radioactive substance decays from 80 grams to 50 grams in 12 years. Find the half-life.
Hint: Find k using A(12) = 50, A₀ = 80. Then set A(t) = 40 (half of 80) and solve for t.
$5000 is invested at 3.2% compounded continuously. What is the balance after 8 years?
Hint: Use A = Pe^(rt) with P = 5000, r = 0.032, t = 8.
A turkey at 165°F is placed in a 70°F room. After 30 minutes it cools to 140°F. Find the temperature after 90 minutes.
Hint: Set up Newton's Law: T(t) = 70 + 95·eᵏᵗ. Use T(30) = 140 to find k, then evaluate T(90).
The logistic model P(t) = 8000/(1 + 15·e^(−0.2t)) models fish in a lake. Find the initial population and the population after 10 years.
Hint: Initial population: P(0). For P(10), substitute t = 10 and simplify.
Key Vocabulary
Exponential Growth Model
A(t) = A₀eᵏᵗ with k > 0. Models quantities that increase at a rate proportional to their current size: bacteria, investments, populations.
Exponential Decay Model
A(t) = A₀eᵏᵗ with k < 0. Models quantities that decrease proportionally: radioactive decay, drug concentration, depreciation.
Continuous Compound Interest
A = Pe^(rt). P = principal, r = annual rate (decimal), t = time in years. The limit of compounding as the frequency approaches infinity.
Half-Life
The time for a decaying quantity to reach half its current value. t½ = ln(2)/|k|. Used in radioactive dating and pharmacology.
Doubling Time
The time for a growing quantity to double. t₂ = ln(2)/k. Rule of 70: t₂ ≈ 70/(annual % rate).
Logistic Growth Model
f(t) = c/(1 + ae^(−bt)). Models growth limited by a carrying capacity c. Produces an S-shaped curve.
Carrying Capacity
The maximum sustainable population or value in a logistic model. The horizontal asymptote as t → ∞.
Newton's Law of Cooling
T(t) = T_s + (T₀ − T_s)eᵏᵗ. Models how an object's temperature approaches the surrounding temperature T_s over time.
Check Your Understanding
Interactive Practice — 5 Questions
A population starts at 1000 and grows to 1500 in 5 years. What is the growth constant k (to four decimal places)?
Carbon-14 has a half-life of 5730 years. What fraction remains after 11460 years?
For the logistic model P(t) = 12000/(1 + 5e^(−0.3t)), what is the carrying capacity?
$10000 is invested at 5% compounded continuously for 6 years. Which expression gives the balance?
A substance decays with k = −0.0347. What is its half-life (to the nearest year)?
Independent Practice
Independent Practice
A city's population was 45,000 in 2010 and 58,000 in 2020. Write an exponential model and predict the population in 2030.
A drug has a half-life of 4 hours. A patient takes a 200 mg dose. How much remains after 10 hours?
$12,000 is invested at 6% compounded continuously. How long until the balance reaches $20,000?
A thermometer reads 5°C and is placed in a 25°C room. After 10 minutes it reads 15°C. Find the temperature after 25 minutes.
The logistic model P(t) = 30000/(1 + 29·e^(−0.4t)) models a social media platform's users (in thousands). Find the initial users, users after 5 years, and the carrying capacity.
Common Mistakes
Using A(t) = A₀·(1 + r)^t instead of A(t) = A₀·eᵏᵗ when the problem says "continuous" growth or decay.
Continuous models use base e. Discrete (annual/monthly) models use (1 + r)^t. Read the problem carefully to identify which applies.
Forgetting to convert the percentage rate to a decimal: using r = 5 instead of r = 0.05 in A = Pe^(rt).
Always convert percent to decimal before substituting. 5% → r = 0.05.
Confusing the carrying capacity with the initial value in a logistic model.
In f(t) = c/(1 + ae^(−bt)), the carrying capacity is c (numerator). The initial value is f(0) = c/(1 + a).
Setting up Newton's Law as T(t) = T₀·eᵏᵗ — forgetting to subtract the ambient temperature.
Newton's Law is T(t) = T_s + (T₀ − T_s)·eᵏᵗ. The object approaches T_s, not zero.
Math Tips
Always identify A₀ (initial value) and the given condition (a second point) before solving for k.
Half-life and doubling time are inverses of each other in structure: both use t = ln(2)/|k|. Memorize this one formula for both.
The Rule of 70: doubling time ≈ 70 ÷ (annual % rate). Quick mental check for finance problems.
In Newton's Law of Cooling, k is always negative (the object cools toward T_s). If you get a positive k, recheck your setup.
For logistic models: P(0) = c/(1 + a). As t → ∞, P → c. These two facts let you quickly sanity-check any logistic answer.