AP PRECALCULUS — UNIT 2B · EXPONENTIAL & LOGARITHMIC FUNCTIONS
2.14 — Logarithmic Function Context and Data Modeling
Notes — Real-World Logarithmic Models
💡 Learning Objectives (2.14.A)
By the end of this lesson you will be able to:
- Construct a logarithmic model from contextual or numerical data
- Use a logarithmic model to make predictions and answer questions in context
- Interpret parameters of a logarithmic model in their real-world setting
- Recognize when a logarithmic model is appropriate by analyzing the rate of change
1. When Is a Logarithmic Model Appropriate?
A quantity follows a logarithmic pattern when equal multiplicative changes in the input produce equal additive changes in the output. Equivalently, the rate of change is positive but decreasing — fast growth at first, then leveling off. Look for situations where:
- The phenomenon is naturally measured on a scale that compresses huge ranges (sound, earthquakes, acidity)
- Each successive ‘doubling’ of input produces the same fixed gain in output
- A graph appears to grow without bound but at an ever-slower rate
2. The General Form
A common general form is y = a + b · log_c(x).
- a is a vertical shift — the output value when log_c(x) = 0, i.e. when x = 1
- b is a vertical scale factor — when b > 0 the curve increases; when b < 0 it decreases
- c is the base — most data models use base 10 or base e for convenience
3. Three Classic Logarithmic Scales
💡 Real-World Logarithmic Scales
- Decibels (sound intensity): L = 10 · log(I/I₀), where I₀ is a reference intensity. A 10-dB increase corresponds to a tenfold increase in sound intensity.
- Richter magnitude: M = log(A/A₀). A magnitude-7 quake has 10× the amplitude of a magnitude-6.
- pH: pH = −log[H⁺]. Acidic solutions have low pH; each pH unit represents a tenfold change in hydrogen-ion concentration.
4. Building a Model from Two Points
If a context gives you two (x, y) data points, you can usually solve for the parameters of y = a + b · log(x).
📘 Example — Fit y = a + b · log(x) to (1, 4) and (100, 10)
- Plug in (1, 4): 4 = a + b · log(1) = a + 0 = a, so a = 4
- Plug in (100, 10): 10 = 4 + b · log(100) = 4 + 2b, so b = 3
- Model: y = 4 + 3 log(x)
- Quick check at x = 10: y = 4 + 3(1) = 7 — between 4 and 10, as the slow growth predicts ✓
5. Using the Model to Make Predictions
📘 Example — Decibel level
- Quiet conversation: I = 10⁻⁹ W/m²; loud rock concert: I = 10⁻¹ W/m². With reference I₀ = 10⁻¹², compute each level.
- Conversation: L = 10 log(10⁻⁹/10⁻¹²) = 10 log(10³) = 30 dB
- Concert: L = 10 log(10⁻¹/10⁻¹²) = 10 log(10¹¹) = 110 dB
- The intensity ratio is 10⁸ — an immense range — but the dB difference is only 80.
6. Interpreting the Parameters
On the AP exam, you may be asked what the value of a parameter ‘means in context.’ The trick is to translate the symbol into a sentence:
- a is the predicted output when the input equals 1 unit (because log(1) = 0)
- b · log(c · x) increases by b every time x is multiplied by c — a constant ‘gain per multiplication’
- Negative b indicates the modeled quantity decreases with bigger inputs
7. Choosing Between Models
How do you decide whether a logarithmic, linear, or exponential model fits best?
- LINEAR if equal additive changes in x produce equal additive changes in y
- EXPONENTIAL if equal additive changes in x produce equal MULTIPLICATIVE changes in y
- LOGARITHMIC if equal MULTIPLICATIVE changes in x produce equal additive changes in y
Logarithmic and exponential are inverses, so the sentence test is symmetric — that is exactly why one is the inverse of the other.
8. Summary
- y = a + b · log_c(x) is the standard form of a logarithmic model
- Logarithmic scales (dB, pH, Richter) compress huge ranges into manageable numbers
- Two data points can determine a and b once the base is chosen
- Interpret parameters in context: a is value at x = 1; b governs ‘gain per multiplication’
- Use the equal-change test to decide which family of model fits the data