Exponential and Logarithmic Functions: Transformations and Properties

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 $y = \log_3(x - 1)$.

• This is a horizontal shift of the parent function $y = \log_3(x)$ to the right by 1 unit.

• The graph of $y = \log_3(x)$ has a vertical asymptote at $x = 0$; after the shift, the asymptote is at $x = 1$.

Domain and Range of Logarithmic Functions

The domain of a logarithmic function $y = \log_b(x - h)$ is all real numbers $x$ such that $x - h > 0$, or $x > h$. The range is all real numbers.

• Domain: $x > 1$ for $y = \log_3(x - 1)$

• Range: $(-\infty, \infty)$

Equation of the Asymptote

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

• For $y = \log_3(x - 1)$, set $x - 1 = 0$ to find the asymptote.

• Equation: $x = 1$

Graphing Logarithmic Functions

To graph $y = \log_3(x - 1)$:

1. Start with the graph of $y = \log_3(x)$.

2. Shift the graph right by 1 unit.

3. Draw the vertical asymptote at $x = 1$.

4. Plot key points, such as $(2, 0)$ since $\log_3(2 - 1) = \log_3(1) = 0$.

Summary Table: Properties of $y = \log_3(x - 1)$

Property	Value
Domain	$x > 1$
Range	$(-\infty, \infty)$
Vertical Asymptote	$x = 1$
Parent Function	$y = \log_3(x)$
Transformation	Shift right by 1 unit

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).