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Logarithmic Models (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Logarithmic models

When is a logarithmic model appropriate?

Logarithmic functions are the inverses of exponential functions
- so they model the 'reverse' type of relationship
An exponential function has
- outputs that change proportionally (multiplicatively)
  - as inputs change additively (by equal amounts)
A logarithmic function has the opposite behavior
- input values change proportionally
  - over equal-length output-value intervals
- In other words, the inputs must be multiplied by a constant factor
  - to produce equal increases in the output
- E.g. the Richter scale for earthquake magnitude is logarithmic
  - Each increase of 1 on the scale corresponds to a tenfold increase in the measured amplitude of the earthquake

How do I construct a logarithmic model from two data points?

A logarithmic model can be constructed from two input-output pairs using the form $f (x) = a + b \ln (x + c)$
- where $c$ is often determined by the context
  - commonly $c = 0$ or $c = 1$
- On the exam, a question typically provides the form of the model
  - e.g. $G (t) = a + b \ln (t + 1)$
  - and asks you to find the constants $a$ and $b$

Here is the general process (using $f (x) = a + b \ln (x + c)$ with data points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ )
- Assume that $c$ is known (it is usually given on the exam)
- Substitute each data point into the function to get two equations
  - $y_{1} = a + b \ln (x_{1} + c)$
  - $y_{2} = a + b \ln (x_{2} + c)$
- Solve the system for $a$ and $b$
- Often one of the data points will cause the argument inside one of the logarithms to be equal to 1
  - This simplifies things greatly since $\ln (1) = 0$
  - It gives $a = y_{1}$ directly

Example: A study uses test scores to track academic content retention. At $t = 0$ months, the average score is 75 points. At $t = 3$ months, the average score is 70.84 points. The model to be used is $R (t) = a + b \ln (t + 1)$ .
- $R (0) = 75$ so
  - $75 = a + b \ln (0 + 1) \Rightarrow 75 = a + b \ln (1) \Rightarrow a = 75$
    - this uses $\ln (1) = 0$ in the final step
- And $R (3) = 70.84$ , so
  - $70.84 = a + b \ln (3 + 1) \Rightarrow 70.84 = 75 + b \ln (4)$
    - using the value of $a$ from the previous step
- Solve for $b$
  - $- 4.16 = b \ln (4) \Rightarrow b = - \frac{4.16}{\ln (4)} \approx - 3.001$
- So the model is
  - $R (t) = 75 - 3.001 \ln (t + 1)$

Examiner Tips and Tricks

Constructing a model from two data points tends to appear in a free response question on every exam, so practice this skill thoroughly.

Non-exact values of $a$ and $b$ must be given as decimal approximations correct to three decimal places
- The scoring guidelines are strict about this
You can solve for $a$ and $b$ either algebraically or by using your graphing calculator
- The scoring guidelines note that "supporting work is not required", so using a calculator is perfectly acceptable
Store intermediate values in your calculator rather than rounding them
- Rounding too early can produce final answers that are not accurate to three decimal places

Can a logarithmic model also be constructed from a proportion and a zero?

Yes, a logarithmic model can be constructed from:
- An appropriate proportion
  - i.e. the multiplicative factor for inputs over equal output intervals
- A real zero
  - an input value where the output is zero
This connects to the inverse relationship with exponential functions
- If an exponential function has base $b$ and initial value 1
  - then its inverse logarithmic function has base $b$ and a zero at $x = 1$
E.g. for a model in the form $f (x) = \log_{b} (k x)$
- suppose input values are multiplied by 2 for each increase by 1 in the output
  - and the output is 0 when $x = 4$
- The proportion, which determines the base of the logarithm, is $b = 2$
  - inputs double per unit output increase
- And the zero is at $x = 4$ , so $k x$ must equal 1 when $x = 4$
  - because $\log_{b} (1) = 0$
- Therefore the model is
  - $f (x) = \log_{2} (\frac{x}{4})$

How can transformations be used to build a logarithmic model?

Logarithmic models can be constructed by applying transformations to the base function $f (x) = \log_{b} x$
The transformations allows the function to match the characteristics of the context or data:
- A vertical shift $\log_{b} (x) + k$
  - adds $k$ to all the output values
    - the graph goes through $(1, k)$ instead of $(1, 0)$
    - the location of the zero will move
- A horizontal shift $\log_{b} (x + h)$
  - shifts the graph by $h$ units to the left
    - the zero will occur at $(1 - h, 0)$ instead of at $(1, 0)$
- A vertical dilation $a \log_{b} x$
  - adjusts the rate of growth

How do I construct a logarithmic model using technology?

When data does not fit a simple logarithmic form exactly
- a graphing calculator can fit a logarithmic regression model to the data
This is done using the LnReg (or similar) function on most graphing calculators
- The calculator determines the best-fit values for the constants $a$ and $b$
- in the model $a + b \ln x$

Why is the natural logarithm often used in modeling?

The natural logarithm ( $\ln$ , with base $e \approx 2.718$ ) appears frequently in real-world models
- Many natural processes involve continuous growth or decay
  - and the natural base $e$ arises naturally in these contexts
In logarithmic modeling questions on the exam, $\ln$ appears far more regularly than other logarithm bases

How can a logarithmic model be used to make predictions?

Once a logarithmic model is constructed, it can be used to predict values of the dependent variable
- by substituting input values into the function
You can also predict input values
- by substituting output values into the function
- and solving to find the corresponding input value
Some things to keep in mind:
- Logarithmic functions grow very slowly for large inputs
  - Predictions for very large input values may not change much
- The model may not remain valid beyond the range of data used to construct it
  - Just as with any model

Worked Example

The number of bird species observed in a nature reserve can be modeled by the function $S$ given by $S (t) = a + b \ln (t + 1)$ , where $S (t)$ is the total number of species observed during the first $t$ months of a long-term survey.

At the start of the survey $(t = 0)$ , 15 species were observed. After 5 months $(t = 5)$ , 24 species had been observed.

(a) Use the given data to write two equations that can be used to find the values for constants $a$ and $b$ in the expression for $S (t)$ .

Answer:

Substitute the two data points into $S (t) = a + b \ln (t + 1)$

$S (0) = 15$
$a + b \ln (0 + 1) = 15$

$a + b \ln (1) = 15$

$S (5) = 24$
$a + b \ln (5 + 1) = 24$

$a + b \ln (6) = 24$

(b) Find the values for $a$ and $b$ as decimal approximations.

Answer:

Use $\ln (1) = 0$ in the first equation

$a = 15$

Substitute that into the second equation

$15 + b \ln (6) = 24$

$b \ln (6) = 9$

$b = \frac{9}{\ln 6} = 5.022995 . . .$

Round to 3 decimal places

$b = 5.023 (3 d . p .)$

Worked Example

Two function models $f$ and $g$ are constructed to represent the depth of water, in meters, in a reservoir after $t$ weeks, for $t \geq 2$ . The functions are given by $f (t) = 16 - 3.6 \ln t$ and $g (t) = - t + 16$ .

What is the first time $t$ that the depth predicted by the logarithmic model will be 0.1 meters more than the depth predicted by the linear model?

(A) $t = 8.713$

(B) $t = 7.213$

(D) $t = 5.713$

Answer:

You need to find when $f (t) - g (t) = 0.1$

$(16 - 3.6 \ln t) - (- t + 16) = 0.1$

$t - 3.6 \ln t = 0.1$

This equation cannot be solved algebraically, so use a graphing calculator

Graph $y_{1} = t - 3.6 \ln t$ and $y_{2} = 0.1$
- then find the intersection (or solve $t - 3.6 \ln t - 0.1 = 0$ ).
The $t$ -value of the intersection is the time you are looking for

Be careful here, as there are actually 2 points of intersection

But only one of them occurs for the interval $t \geq 2$

Graph of a curve intersecting y=0.1 at labelled point (7.21337, 0.1), on a grid with marked axes.

To 3 decimal places that $t$ -coordinate has value 7.213

(B) $t = 7.213$

Examiner Tips and Tricks

The second Worked Example requires using your graphing calculator because the equation mixes a polynomial term ( $t$ ) with a logarithmic term ( $\ln t$ ).

Recognizing when technology is needed (rather than spending time trying to solve algebraically) is an important exam skill

Also note that you may also be able to solve $t - 3.6 \ln t = 0.1$ using your calculator's equation solving functionality.

This is an alternative to drawing a graph and finding the intersection
You would still need to take care to find the solution that satisfies $t \geq 2$

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Logarithmic Models (College Board AP® Precalculus): Study Guide

Logarithmic models

When is a logarithmic model appropriate?

How do I construct a logarithmic model from two data points?

Examiner Tips and Tricks

Can a logarithmic model also be constructed from a proportion and a zero?

How can transformations be used to build a logarithmic model?

How do I construct a logarithmic model using technology?

Why is the natural logarithm often used in modeling?

How can a logarithmic model be used to make predictions?

Worked Example

Worked Example

Examiner Tips and Tricks

Unlock more, it's free!

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Author: Roger B

Reviewer: Mark Curtis