AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/2/2-14)
Unit 2 · Topic 2.14
2.14 Logarithmic Function Context and Data Modeling
Logarithmic models fit situations where a quantity rises quickly at first and then more and more slowly, because equal multiplications of the input give equal additions to the output. You'll build log models from context, from two points, or by regression, and use them in both directions.
Key terms
- logarithmic model
- logarithmic regression
- context
- interpretation
When a log model fits
Use a log model when the inputs change proportionally (each step multiplies the input by the same factor) while the outputs change by equal amounts.
Another way to see it: a log counts multiplications. If the output is a whole number, it tells you how many times the starting value has been multiplied by the base to reach the input. log₂(64) = 6 says 1 has to be doubled 6 times to reach 64.
Real examples include the decibel scale for loudness, the pH scale in chemistry, the magnitude scale for earthquakes, and learning curves where each new improvement takes much more practice than the last.
Compare it with an exponential model. Exponential: add to the input, multiply the output. Logarithmic: multiply the input, add to the output.
Building a log model
From a proportion and a zero: if the output is 0 at x = x₀ and rises by c every time x is multiplied by k, then y = c · log_k(x/x₀).
From two points: use the form y = a + b · ln x (or a + b · log_k x) and solve the two equations for a and b.
From transformations: start with log_b x and apply shifts and dilations to match the context.
From data: logarithmic regression on a calculator returns y = a + b ln x.
Example of the first method: if a quantity is 0 when x = 5 and rises by 4 every time x triples, then y = 4 log₃(x/5).
The natural log in models
Calculators give logarithmic regression in terms of ln. Any log model can be written this way, because by change of base, log_k x = ln x / ln k, which is just ln x times a constant.
Using a log model
Predict an output by plugging in x. Find the input that gives a target output by solving the log equation, which means rewriting in exponential form.
You won't be asked to interpret the constants a and b of a log model like y = a + b ln x in context; that's outside the course. It still helps to see what they do: in y = 7 + 3 log₂ x, every doubling of x adds 3 to y, and y = 7 when x = 1.
As with any model, check the domain. Log models need positive inputs, and predictions far outside the data may not make sense.
Logs grow without bound, so a log model can overshoot a quantity that really levels off. A test score can't go past 100, but a log model of it keeps rising forever. Use the model only over a sensible range of inputs.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
A log model from two points
In a video game, a player's level L after x hours of play is modeled by L(x) = a + b log₂(x). A player is at level 10 after 2 hours and level 16 after 8 hours. Find a and b, predict the level after 32 hours, and find when the player reaches level 25.
Show the solutionHide the solution
- Step 1: Use the points: L(2) = a + b log₂ 2 = a + b = 10, and L(8) = a + b log₂ 8 = a + 3b = 16.
- Step 2: Subtract the first equation from the second: 2b = 6, so b = 3. Then a = 7.
- Step 3: L(x) = 7 + 3 log₂(x). Check: L(2) = 7 + 3 = 10 and L(8) = 7 + 9 = 16.
- Step 4: L(32) = 7 + 3 · 5 = 22.
- Step 5: Level 25: 7 + 3 log₂ x = 25, so log₂ x = 6 and x = 2⁶ = 64.
Answer: L(x) = 7 + 3 log₂(x); level 22 after 32 hours; level 25 after 64 hours.
- Example 2Calculator allowed
Logarithmic regression
A skill score y after x practice sessions is recorded: (1, 2.1), (2, 4.9), (4, 8.2), (8, 10.8), (16, 14.1), (32, 16.9). Find a logarithmic regression model, predict the score after 20 sessions, and find when the model predicts a score of 12.
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- Step 1: The inputs double while the outputs rise by about 3 each time: the log pattern.
- Step 2: Logarithmic regression gives y ≈ 2.057 + 4.295 ln(x).
- Step 3: y(20) ≈ 2.057 + 4.295 ln 20 ≈ 14.924.
- Step 4: Solve 2.057 + 4.295 ln x = 12 using the stored values: ln x ≈ 2.3149, so x ≈ e^2.3149 ≈ 10.124.
Answer: y ≈ 2.057 + 4.295 ln x; about 14.924 after 20 sessions; a score of 12 at about 10.124 sessions (so 11 is the first whole number of sessions where the model predicts more than 12).
- Example 3
A model from a proportion and a zero
Sound level in decibels is L = 10 log(I/I₀), where I is the sound's intensity and I₀ is the faintest sound a person can hear. How much louder, in decibels, is a sound with 1,000 times the intensity of another?
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- Step 1: When I = I₀, L = 10 log 1 = 0: that's the zero of the model.
- Step 2: Multiplying the intensity by 10 adds 10 log 10 = 10 decibels.
- Step 3: Multiplying by 1,000 = 10³ adds 10 log(1000) = 10 · 3 = 30 decibels, by the product property.
Answer: 30 decibels louder.
Common mistakes
- Using a log model for data whose inputs are evenly spaced and whose outputs multiply. That's an exponential pattern, the reverse of a log pattern.
- Plugging a target output into the model as if it were an input. To find x from y, solve the log equation.
- Rounding regression constants before solving for x, which can shift the answer noticeably.
On the exam
- On the non-periodic modeling free-response question, a log model may appear with constants to find from given points. Show the system of equations.
- To find the input that gives a target output, rewrite the log equation in exponential form and show it, as in log₂ x = 6, so x = 2⁶ = 64.
Connected topics
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Check yourself
4 questions on 2.14 Logarithmic Function Context and Data Modeling. Pick an answer to see if you got it, and why.
The height of a seedling, in centimeters, is modeled by h(t) = 2 + 3 ln(t + 1), where t is the number of days after the seedling sprouts. According to the model, how many days after sprouting does the seedling first reach a height of 8 centimeters?
| d (days) | 1 | 2 | 4 | 8 | 16 |
|---|---|---|---|---|---|
| N (hundreds) | 3.1 | 5.0 | 7.2 | 9.0 | 11.1 |
Invented data
A logarithmic regression of the form N = a + b ln d is used to model the data. According to the model, which of the following is closest to the number of people, in hundreds, who had heard about the event 30 days after it was announced?
The loudness of a sound, in decibels, is modeled by L = 10 log₁₀(I/I₀), where I is the intensity of the sound and I₀ is a fixed reference intensity. If the intensity of a sound is multiplied by 100, how does its loudness change?
| x | 1 | 3 | 9 | 27 |
|---|---|---|---|---|
| h(x) | 5 | 7 | 9 | 11 |
Table of values
Which of the following could define h?
0 of 4 answered