Describe key features (domain, range, asymptotes, concavity, and end behavior) of the graph of a parent logarithmic function, y=log_b(x).
Sketch parent logarithmic functions and their transformations.
Connect key features on the graphs of exponential and logarithmic functions.
Quick Lesson Plan
This lesson introduces graphs of logarithmic functions, which are the inverses of exponential functions. Students should work through Questions 1-5 in groups to explore graphs of logarithms. They start by exploring the graph of a log with base 2, where they’ll identify the domain, range, and see the vertical asymptote at x = 0.
In Question 4, we want them to see how the base of a log affects the steepness of the curve, just like with exponential functions, but does not change the x-intercept or vertical asymptote. They will then use their knowledge of transformations to identify the changes in a graph that has been transformed 3 units to the right.
Why does the graph [of log base 2] go through (1,0)?
Why does the graph go through (2,1)?
Why is the domain [of log base 2] only positive numbers?
What does a negative output mean here?
How can you tell what the base of a logarithmic function is from the graph?
What is the concavity of this graph? What does that tell you about the rate of change of the function?
Remind the students of the connection between logarithmic and exponential functions. In Question 4, emphasize that all parent logarithmic functions go through the point (1, 0) because log(1) = 0 and through the “base point” (b, 1) since b^1=b which means log base b of b is 1. They will see that transformations affect the x-intercept, asymptote, and domain, but not the range. In fact, no transformations will affect the range since the range of any logarithmic function is always all real numbers.
In question 1 of the in the Check Your Understanding, students will see a reflection across the y-axis that will affect the domain of the function, but keep the vertical asymptote the same. Emphasize that you cannot evaluate the log of 0 or the log of a negative number.
In question 3 of the Check Your Understanding, students construct a logarithmic function that satisfies two criteria. Students can approach this from a graphical perspective (what transformations occurred) or an algebraic perspective (solving a system of equations to determine the parameters). Note that there could be more than one right answer that satisfies the given criteria.
One tricky aspect of logarithmic graphs is that if there is a horizontal shift or vertical shift, the x-intercept is no longer (1,0) and the point (b,1) is not on the graph. This makes it much more challenging for students to identify the base of the logarithm. In this case it may be advantageous to use an algebraic approach to determining the parameters in the equation.