Exponential and Logarithmic Functions

Transformations of Logarithmic Functions

Logarithmic functions are a key topic in College Algebra, often explored through their graphs and transformations. The general form of a logarithmic function is y = log_b(x), where b is the base of the logarithm.

• Vertical and Horizontal Shifts: Adding or subtracting a constant inside the logarithm shifts the graph horizontally, while adding or subtracting outside the logarithm shifts it vertically.

• Reflections: Multiplying the function by -1 reflects the graph across the x-axis.

• Stretching and Compressing: Multiplying the input or output by a constant stretches or compresses the graph.

Example: Consider the function .

• This is a horizontal shift of the parent function to the right by 1 unit.

• The graph of has a vertical asymptote at ; after the shift, the asymptote is at .

Domain and Range of Logarithmic Functions

The domain of a logarithmic function is all real numbers such that , or . The range is all real numbers.

• Domain: for

• Range:

Equation of the Asymptote

Logarithmic functions have a vertical asymptote where the argument of the logarithm is zero.

• For , set to find the asymptote.

• Equation:

Graphing Logarithmic Functions

To graph :

1. Start with the graph of .

2. Shift the graph right by 1 unit.

3. Draw the vertical asymptote at .

4. Plot key points, such as since .

Summary Table: Properties of

Additional info: The original file is a review exam question focusing on graph transformations, domain/range, and asymptotes for logarithmic functions, which are central topics in College Algebra (Ch. 4).