Exploring Logarithmic Models

Students explore the Countries of the World Starter File, and find a relationship between wealth and median life expectancy. They also learn how to build a new column for a table in Pyret.

Direct students to look at the scatter plot they created in Pyret of median-lifespan versus pc-gdp.

Having identified that the wealth-v-lifespan scatter plot is neither linear, nor quadratic, nor exponential, students learn about characteristics of logarithmic functions in graphical, tabular, and function notation form.

Have students share their answers. For logarithmic functions, it’s especially important to have students talk about how much 𝑥 needs to increase just to get a fixed increase in 𝑦. This foreshadows the idea of base for logarithmic relationships.

Logarithmic relationships - like the ones you identified on tables 1, 4, and 6 - grow or decay very quickly at first, but then level out as 𝑥 increases. Our intervals need to grow longer and longer just to reach a small change in 𝑦. For example, it might take 10 steps to go from 1 to 2, but then a hundred more to get to 3 and a thousand more to get to 4.

Logarithmic models have the form 𝑓(𝑥) = 𝑎 log_𝑏(𝑥) + 𝑐

Review students answers, and then debrief via class discussion. Invite students to consider what new information they have gained by looking at graphical representations rather than tables.

The base 𝑏 of the logarithm is similar to the base in exponential functions.

Look back to your answers on What Kind of Model? (Tables). For tables that represent logarithmic functions, by what factor does 𝑥 need to grow in order to get a constant increase in 𝑦?

, (Don’t forget — anything to the power of zero is always 1, so the log of 1 will always be zero - for any base!)

The term 𝑐 is the vertical shift of the function, which moves the curve up or down.
(We’ve seen vertical shifts in other kinds of functions given different names, like 𝑘 for quadratics and 𝑏 for linear function.)

Because log_anything(1) = 0

The term 𝑎 is called the logarithmic coefficient, which - like 𝑏 - determines how quickly the function grows. Extremely observant students may notice that there’s a relationship between 𝑎 and 𝑏, where the value of 2 log₁₀(10) = log₁₀(10²)!

Logarithmic models have a vertical asymptote where the function increases or decreases boundlessly. In this data exploration, the asymptote will always be located on the y-axis (𝑥 = 0).

We measure sound intensity on a logarithmic scale, which proceeds in multiples of 10. The table to the right gives some intensity levels in watts per square meter and in decibels. Our ears can hear incredibly quiet sounds (like a pin dropping), but also process incredibly loud sounds (like a fog horn). A fire alarm, for example, is thousands of times louder than a dog barking, but it’s difficult for our brains to process that much more "loudness". As a result, we also perceive loudness on a logarithmic scale: for us to perceive a sound as being twice as loud as another, it actually has to be a hundred times as loud.

Have students share their answers. Be especially attentive to students who mis-label logarithmic relationships as "exponential" — the relationship between the two is extremely subtle!

Students discover that when a logarithmic relationship is graphed on an exponential scale, the point cloud appears linear. When trying to use linear regression with those points, however, they are reminded that merely changing the scale of a graph does not actually change the data.

This section builds the foundation for linearization, transforming the points themselves, which students will do in the following section. (Note: this also opens the door for teaching inverse functions!)

Remember that logarithmic models have the form 𝑓(𝑥) = 𝑎 log_𝑏(𝑥) + 𝑐

To fit our logarithmic model, we need to find 𝑎 and 𝑐, such that the model fits the data as closely as possible.

Trial-and-error only gets us so far, and it’s not clear that we would ever stumble upon the optimal model. We need something like Pyret’s lr-plot function, which uses computational methods to find the best possible model. Unfortunately, lr-plot only finds linear models!

If only we could transform this data to make it appear linear. Then we could use lr-plot to fit the optimal model, and then reverse the transformation to get the optimal logarithmic model!

Imagine that the scatter plot is printed on a sheet of rubber, and can be stretched or squashed in any way we want. Data Scientists often use transformations to stretch their data into shapes that are easier to use, and then reverse the transformation when they are done.

By transforming the x-axis to grow exponentially, we are "squashing" the coordinate plane so that each interval on the x-axis represents 10x the growth in pc-gdp as the one before it. This balances out the logarithmic growth in median-lifespan, makes the curved relationship appear linear, and warps our logarithmic model so it looks like a straight line-of-best-fit.

The following metaphor might help students make sense of this.

Having discovered that changing the scale of a graph does not allow them to use linear regression, students learn to transform the data into a linear shape, building a new column by applying a function to each row. This new data can be fit with a linear model. By then applying the inverse of this transformation to their computed linear model, they retrieve the logarithmic model.

We tried changing the scale on the x-axis from linear to exponential, which cancels out the logarithmic behavior by "shrinking the axis". Another strategy is to "shrink the data", by transforming the x-coordinates themselves. Instead of plotting pc-gdp on a logarithmic x-axis, we could plot log(gdp) on a linear x-axis.

Transforming the points instead of the axis has the same visual effect: the dots appear to fall in a straight line. But now we can plot them on a linear-scale axis, and use linear regression to find the best-possible model!

This transformation changed the kind of growth from logarithmic to linear: the term went from 𝑎 log₁₀(𝑥) to 𝑎𝑥. Instead of increasing logarithmically by 𝑎, our new function increases linearly by 𝑎.

We transformed the pc-gdp column in three steps:

It’s easy to do the same thing in Pyret!

Address any student questions about the Pyret function they’ve just discovered, build-column. Verify that students have recorded the slope and vertical shift for their regression line. Then, emphasize the key ideas below.

Just like in Desmos, transforming the pc-gdp column with a log function produces a scatter plot showing a linear pattern in the data! Pyret’s lr-plot tool computes the best possible linear model for our transformed data, determining it to have a slope of 11.9011 and a y-intercept of 24.2636. Our 𝑅² has jumped to 0.66311, showing a vastly better correlation than before.

With the transformation applied, our linear model (in both function and Pyret notation) is:

From Transforming the Data, we know that the coefficients used in the transformed, linear model are the same ones used in the logarithmic, un-transformed model:

The resulting logarithmic model can be fit to our original scatter plot, showing a much better fit than our 2-point-derived estimates.