Graphing Logarithmic Functions

This section of the lesson focuses on graphing logarithmic functions and understanding their relationship to exponential functions.

The lesson explains that since f$x$ = log_b x and g$x$ = b^x are inverse functions, their graphs are reflections of each other across the line y = x.

Highlight: The y-axis is a vertical asymptote for the graph of f$x$ = log_b x.

Key points about logarithmic graphs are presented:

• Domain: x > 0
• Range: all real numbers
• Vertical asymptote at x = 0

Example: Graph f$x$ = log_2$x-1$ This example demonstrates how to graph a transformed logarithmic function.

The lesson provides a graphing strategy:

1. Rewrite the function in exponential form
2. Create a table of values
3. Plot the points and draw the curve

Vocabulary: Asymptote - a line that a curve approaches but never touches

Parent Graphs and Transformations of Logarithmic Functions

This section introduces the parent graphs for logarithmic functions and explores how to graph transformed logarithmic functions.

The lesson presents two cases:

1. Graph of f$x$ = log_b x for b > 1
2. Graph of f$x$ = log_b x for 0 < b < 1

Highlight: The graph of f$x$ = log_b x is the reflection of g$x$ = b^x in the line y = x.

The lesson then moves on to graphing transformed logarithmic functions. It provides an example of graphing f$x$ = log_3$x-1$ and explains the steps involved.

Example: Given y = a log$x-h$ + k, the vertical asymptote is at x = h

Students are then given an opportunity to practice graphing a transformed logarithmic function:

Example: Graph f$x$ = -log_2 x - 1

The lesson concludes with a formative assessment, asking students to rewrite and graph y = log_2 x - 1.

Vocabulary: Transformation - a change in the shape, size, or position of a graph

This comprehensive lesson provides students with a solid foundation in logarithmic function graphs, their relationship to exponential functions, and how to apply transformations to these graphs.

6.3 Day 2 Lesson: Logarithmic Functions

This lesson begins with a review of logarithmic expressions and their simplification. Students are reminded of an upcoming graded assessment and are instructed to check their answers for previous practice activities.

Highlight: The lesson emphasizes the inverse relationship between exponential and logarithmic functions.

The lesson introduces the concept that logarithmic functions are the inverse of exponential functions. This is expressed mathematically as:

g(f$x$) = log b^x = x f(g$x$) = b^$log_b x$ = x

Definition: Logarithmic functions and exponential functions "undo" each other, meaning they are inverse functions.

Students are then given practice problems to simplify logarithmic expressions and evaluate logarithms.

Example: Simplify log_5$5^2$ = 2

The lesson also covers finding the inverse of functions, including exponential and logarithmic functions.

Vocabulary: The inverse function is denoted as f^-1$$x and represents the "undoing" of the original function.