Topic 2.14 Logarithmic Function Context and Data Modeling: construct a logarithmic model from a context or data set, interpret its parameters, and use it to make predictions.

What this topic is asking

The College Board (Topic 2.14) wants you to construct a logarithmic model from a context or data set, interpret its parameters, and use it to predict. A logarithmic model fits data that rise quickly then level off, with a slowing rate of increase. You should build a model from a context, fit data with logarithmic regression, and interpret real logarithmic scales such as pH, decibels and the Richter scale.

When a logarithmic model fits

The clue in a context is a quantity that grows fast at small inputs and then gives diminishing returns, or a scale that responds to ratios rather than absolute differences.

Logarithmic scales

These scales are the most common contexts for logarithmic modelling, and the exam tests interpreting "each unit means times ten".

Building and predicting

To build a logarithmic model from data on the calculator section, enter the points and run logarithmic regression, which returns $a$ and $b$ in $a + b\ln x$. Then predict by substituting, and interpret the coefficient $b$ as how strongly the output responds to multiplicative changes in the input. As with exponential modelling (Topic 2.5), you validate the choice using residuals (Topic 2.6) before trusting predictions.

A point that earns marks is articulating the "diminishing returns" behavior in context. A logarithmic model predicts that doubling the input does not double the output; instead the output rises by a fixed amount each time the input multiplies by a fixed factor. Saying explicitly that "each tenfold increase adds a constant" or "later gains require ever larger inputs" demonstrates that you understand why a logarithmic model, rather than a linear or exponential one, suits the situation, which is the interpretive depth the free-response rubric looks for.

A clarifying idea is that logarithmic and exponential models are inverses in shape as well as in algebra. If a quantity grows exponentially in time, then time expressed as a function of that quantity grows logarithmically. So a logarithmic model often appears when the roles of input and output in an exponential situation are swapped, which is exactly why the same contexts (growth, intensity, concentration) generate both kinds of model depending on which variable you solve for.

Try this

Q1. A sound's loudness is $L = 10\log\left(\frac{I}{I_0}\right)$. By how much does $L$ change if $I$ increases by a factor of $100$? [1 point]

• Cue. $\log(100) = 2$, so $L$ increases by $10 \cdot 2 = 20$ decibels.

Q2. Data rises steeply at small $x$ and flattens as $x$ grows. Logarithmic or exponential model? [1 point]

• Cue. Logarithmic: fast then flattening (decelerating), unlike exponential growth which accelerates.