Section 4.7

Exponential & Logarithmic Models

Word problems using these models are a final exam staple. They combine the equation-solving skills from section 4.6 with real-world context — expect at least one growth/decay or half-life problem.

⚡ Quick Summary

Almost every real-world growth or decay situation follows the same formula: $N = N_0 e^{kt}$ . Think of $N_0$ as where you start, $k$ as the speed dial (positive = growing, negative = shrinking), and $t$ as time. Bacteria multiplying, radioactive material breaking down, money earning interest, a hot cup of coffee cooling — they all follow this same pattern. Once you learn one, you've learned them all.

Exponential Growth/Decay Model

N(t) = N_0 \, e^{kt}

k > 0 for growth, k < 0 for decay

Doubling Time

t_{\text{double}} = \frac{\ln 2}{k}

Time for a quantity to double (assumes k > 0)

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

Word problems in this section almost always follow a three-step recipe: (1) write down the model and fill in what you know, (2) use a given data point to solve for $k$ (the growth/decay rate), and (3) plug $k$ back in to answer whatever the question actually asks. The trick is recognizing that step 2 always comes first — you need to find $k$ before you can predict anything.

Two useful shortcuts: doubling time tells you how long it takes something to double ( $t = \frac{\ln 2}{k}$ ), and half-life tells you how long it takes to lose half ( $t = \frac{\ln 2}{|k|}$ ). These come straight from the main formula — they're just what happens when you set $N = 2N_0$ or $N = \frac{1}{2}N_0$ and solve for $t$ .

Newton's law of cooling is a slight twist: a hot (or cold) object approaches the surrounding temperature over time, not zero. The formula is $T(t) = T_s + (T_0 - T_s)e^{kt}$ , where $T_s$ is the room temperature. Think of it this way — your coffee doesn't cool to absolute zero; it cools to room temperature and stops. Logistic growth is another twist: the population levels off at a maximum (carrying capacity) instead of growing forever, like bacteria that run out of food in a petri dish.

Key Formulas

Exponential Growth/Decay Model

N(t) = N_0 \, e^{kt}

k > 0 for growth, k < 0 for decay

Doubling Time

t_{\text{double}} = \frac{\ln 2}{k}

Time for a quantity to double (assumes k > 0)

Half-Life

t_{1/2} = \frac{\ln 2}{|k|}

Time for a quantity to halve (assumes k < 0)

Newton's Law of Cooling

T(t) = T_s + (T_0 - T_s)\,e^{kt}

T_s = surrounding temp, T_0 = initial temp, k < 0

Logistic Growth Model

P(t) = \frac{c}{1 + a\,e^{-bt}}

c = carrying capacity; growth slows as P approaches c

Key Takeaways

✓The continuous growth/decay model $N = N_0 e^{kt}$ : use $k > 0$ for growth, $k < 0$ for decay
✓Doubling time = $\ln 2 / k$ ; half-life = $\ln 2 / |k|$ — both derived from the same model
✓Word problem strategy: use the given data point to find $k$ first, then answer the question with $k$
✓In Newton's law of cooling, $T_s$ is the surrounding temperature — the object's temp approaches $T_s$ over time

⚠️ Common Mistakes

✗Using a positive $k$ for decay problems — if something is shrinking, $k$ must be negative
✗Plugging in the percentage directly instead of a decimal: $6\%$ growth means $k = 0.06$ , not $k = 6$
✗Confusing the half-life shortcut $t = \frac{\ln 2}{|k|}$ with the general model — the shortcut only gives you the half-life itself, not the amount remaining
✗Forgetting that $T_s$ in Newton's law of cooling is the surrounding temperature, not the starting temperature

Builds on: 4.6

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4.6 Exponential & Logarithmic Equations