Understand that a logarithmic model takes quantities that grow proportionally and assigns them output values that grow linearly.
Identify situations that could be modeled with a logarithmic function.
Construct logarithmic models using input-output pairs or transformations.
Use logarithmic function models to predict values of the dependent variable.
Quick Lesson Plan
In today’s lesson, students consider a common scenario that is actually built around logarithms: how many “figures” a salary has! They will notice that a salary has to increase ten-fold to have an extra “figure” because the number of figures is related to the number of digits which is rooted in our base 10 number system. In question 3b and 3c students have to articulate this relationship between a salary in dollars and a salary in figures. Because the figure numbers go up by 1 in the table, students may assume by default a linear relationship. However, it is easy to see that the inputs are not changing in equal intervals.
In question 3d students must describe the relationship with an equation. We purposefully do not dictate which variable is the independent variable and which is the dependent variable. As you are monitoring groups, look for groups that used an exponential equation and groups that used a logarithmic equation. You will want to make sure both appear in the debrief. See notes in the “Formalize Later” section for this key part of the debrief.
Be prepared for students to struggle with question 4 (that’s why we saved it for last!). The “adder’s mindset” is so strong that it is hard for even us as adults to think of “half-way between” as meaning anything other than the average of the two numbers. Thinking of exponents (i.e. logarithms) as the number of multiplications can be very helpful for students. If my input is multiplied by 10 after 2 steps, what happens after the 1st step? Our knee-jerk reaction is to say the input was multiplied by 5, but this isn’t true. Half-way to multiplication by 10 is multiplication by the square root of 10, since multiplying by the square root of 10 twice is the same as multiplying by 10 once.
What is a 2-figure salary? A 1-figure salary?
Can you write the equation using exponents? Can you write the equation using a logarithm?
What does the (common) log of a number represent? (Use powers of 10 as examples)
Why is the figure number 1 more than the log of d?
How many multiplications by 10 occurred?
Is a 6.5 figure salary closer to 100,000 or 1,000,000?
Students may assume that because the inputs grow by a factor of 10 and the outputs increase by 1 that the equation is f=log(d). This is the distinction between how the dependent variable changes with respect to the independent variable, and the actual value of the dependent variable. It is also valuable for students to notice how the transformations that occurred to fit the data differ in the exponential and logarithmic model. Because of how exponents work, a shift to the right of an exponential function represents a vertical shift in its inverse function! For a salary of d dollars, the value of f representing the number of figures must be 1 more than the number of zeros in d.
In the QuickNotes we outline some of the different avenues from which we can construct a logarithmic model. Note that there are assumably less real-world phenomenon that exhibit logarithmic growth than exponential growth (at least according to our google searches) but a key application of logarithmic functions is simply to determine the independent variable value in an exponential relationship, i.e. constructing an inverse function. The first question in the Check Your Understanding demonstrates this approach as students write an equation for G(h) and H(g). Question 2 highlights the algebraic approach of using two input-output pairs to determine the parameters of a logarithmic equation.
Recall from our lesson on constructing models that there are three types of function models. In this lesson we focus on deterministic models, though logarithmic regression models do exist and can be constructed for bivariate data sets. If you’re looking for a data set that could be modeled with logarithmic regression, check out the data on life expectancy and GDP per capita from Our World in Data.