For the illustrated unshifted model, doubling a positive argument adds . For shifted models, compare ratios of the log argument, not raw input.
What a strong response shows
Compare multiplicative input steps.
For an unshifted input log, equal input ratios give equal output differences.
Check the positive argument and residual pattern before predicting.
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What You Need to Know
A logarithmic model has constant additive output changes over equal multiplicative changes of its positive argument. For an unshifted input this argument is ; for a shifted input it is . The outside scale and base determine whether outputs increase or decrease.
Compare positive logarithm arguments by equal ratios, then inspect the corresponding output differences. Check the direction instead of assuming every logarithmic model increases.
If doubling the input adds about the same amount to the output, a logarithmic model may fit.
Using a logarithmic model when the output changes by a constant percent instead.
What kind of input change should you compare for a logarithmic pattern?
Learning Objectives
2.14.A — Construct a logarithmic function model.
Match the prompt verb to the representation it asks you to build or interpret.
. Name the given quantities. . Choose the matching function or representation. . Show the parameter or operation that changes. . Check one value in context.
Starting calculations before identifying what the input and output mean.
Can you state what the input, output, and requested evidence are before choosing a formula?
Key Concepts
A model can use a shifted and scaled logarithm, with an input domain determined by the context.
Choose the input shift so the logarithm’s argument is positive over the observed data.
Use a data point to set the vertical scale and shift after selecting a plausible logarithmic shape.
Ignoring the input restriction created by the logarithm’s argument.
What must be true of the model’s logarithm argument?
Relevant Representations
Context, scatter plot, equation, and residuals reveal whether the output slows in the expected way.
Check the shape and the units before interpreting coefficients.
| Evidence | Look for | Caution |
|---|---|---|
| Context | diminishing gains | units matter |
| Graph | steep then flatter | domain restriction |
| Equation | log input term | shift may be needed |
| Residuals | small and pattern-free | avoid far extrapolation |
Plot the data, fit a logarithmic curve, inspect residuals, and state the prediction only within the modeled input range.
Extrapolating toward a forbidden input or treating the model as linear everywhere.
What visual pattern supports a logarithmic model?
Essential Relationships or Formulas
Use an exact relationship to expose the pattern before substituting values.
Keep the variables and restrictions visible, then substitute with units attached.
Write the relationship, identify each parameter, substitute the known values, and check one output.
Using a familiar formula without checking its domain, base, interval, or unit assumptions.
Which symbol or parameter carries the feature the prompt asks you to find?
Worked Examples
A short, labeled chain of steps is easier to check than a page of unexplained algebra.
Write the representation first, calculate second, and interpret last.
Plot the data, choose a shifted logarithm whose argument is positive, fit the parameters, and inspect residuals.
Skipping the interpretation step and reporting an answer with no units, domain, or model evidence.
Which step would you show so another student could reproduce the result?
Common Mistakes
Most errors come from a mismatch between the pattern, the operation, and the allowed input.
Before submitting, check pattern, operation, domain, and units in that order.
Circle the first line where the pattern or restriction changes. Correct that line, then recompute only the affected steps.
Trusting a calculator output that hides a wrong model, sign, base, or domain.
What independent check would catch your most likely error?
AP Exam Tips
Use the topic workflow to connect the representation, the equation, and the context before calculating.
Read the prompt, choose the smallest reliable representation, and verify the result in context.
. Identify the pattern. . Write the matching form. . Show the key substitution or transformation. . Check units, domain, and direction.
Submitting a number without explaining why the selected model or transformation fits.
Can you name the evidence that supports your final model or answer?
Open this topic’s practice set to apply the method. Work independently, then compare the evidence in your reasoning with the checked explanation.
Can you transfer the method?
For the model , predict at and state the algebraic domain.
Compare your reasoning
The prediction is . The algebraic domain is ; a contextual model may have a narrower useful domain.
Work independently, then compare the mathematical evidence in your answer.