Exponential and Logarithmic Functions

Introduction to Exponential and Logarithmic Functions

Exponential and logarithmic functions are fundamental in College Algebra, providing tools for modeling growth, decay, and solving equations involving exponents and logarithms. Understanding their properties, graphs, and transformations is essential for analyzing mathematical relationships.

Exponential Functions

Definition and Properties

• Exponential Function: A function of the form , where and .

• Domain:

• Range:

• Y-intercept:

• Horizontal Asymptote:

Example: Find the base if the graph of contains the point .

• Substitute:

• Solve for :

Inverse of Exponential Functions

• The inverse of is .

• To find the inverse, switch and and solve for :

Logarithmic Functions

Definition and Properties

• Logarithmic Function: , where , .

• Domain:

• Range:

• X-intercept:

• Vertical Asymptote:

Example:

Relationship Between Exponential and Logarithmic Functions

• Exponential and logarithmic functions are inverses of each other.

• If , then .

• If , then .

Properties of Logarithms

Examples of Equivalent Logarithmic Expressions

Graphing Exponential and Logarithmic Functions

Graphing and

• Table of values can be used to plot both functions.

• The graphs are reflections of each other across the line .

Table Example:

Characteristics of Inverse Functions

Note: The graph of and are reflections over the line .

Transformations of Logarithmic Functions

General Form

All transformations of the parent logarithmic function have the form:

• Shift horizontally by units:

• Shift vertically by units:

• Reflect about the x-axis:

• Stretch/compress vertically by a factor of :

For , the graph is reflected about the y-axis.

Order of Transformations

• Start with horizontal shift (H)

• Apply any stretching or reflecting (S, R)

• Finish with vertical shift (V)

Example of Transformations

• Given , describe the transformations for :

• H: Shift right

• S: Stretch vertically by a factor of 2

• R: Reflect about y-axis and x-axis

• V: Shift up 2 units

Graphing Transformed Logarithmic Functions

Examples

• For :

• Domain:

• Range:

• X-intercept:

• Asymptote:

Inverse of Logarithmic Functions

Finding the Inverse

• Given , solve for in terms of :

• Domain of :

• Range of :

• Y-intercept:

• Asymptote:

Practice Problems

Sample Questions

1. For , find:

   • Domain:

   • Range:

   • Y-intercept:

   • Equation of the asymptote:

   • Inverse:

2. For , find:

   • Domain:

   • Range:

   • X-intercept:

   • Equation of the asymptote:

   • Inverse:

Summary Table: Exponential vs. Logarithmic Functions

Additional info: The notes include step-by-step examples, graphical sketches, and practice problems to reinforce understanding of logarithmic and exponential functions, their inverses, and transformations. Students are encouraged to use properties of logarithms and exponentials to solve equations and analyze graphs.