A. Use transformations of the graphs of y=log2 x (see...

a. Use transformations of the graphs of $y=\log _{2} x$ (see Example 7) and $y=\log _{3} x$ (see Exercise 69) to graph the given functions.
b. Write the domain and range in interval notation.
c. Determine the vertical asymptote.
$$
y=\log _{3}(x-1)-3
$$

Key Concepts

Graph Transformations

Graph transformations involve taking a basic function and modifying its appearance through translations (shifts), dilations (stretching or compressing), and reflections. In the context of logarithmic functions, horizontal shifts (like replacing x with (x - h)) and vertical shifts (such as subtracting a constant from the function) alter the graph's position, including critical features such as intercepts and asymptotes.

Logarithmic Functions

Logarithmic functions are the inverse functions of exponential functions, defined only for positive arguments. They exhibit a characteristic slow increasing behavior and have distinct properties such as a vertical asymptote where the function is undefined, making them applicable in many real-world situations involving exponential growth or decay.

Domain and Range

The domain of a function is the set of input values for which the function is defined, while the range is the set of possible output values. For logarithmic functions, the domain is determined by the condition that the argument of the logarithm must be positive, and the range often comprises all real numbers, with these intervals often being expressed in interval notation.

Vertical Asymptote

A vertical asymptote is a line that the graph of a function approaches but never reaches; it occurs in logarithmic functions where the argument of the logarithm equals zero. Identifying the vertical asymptote involves finding the value of x that results in an undefined expression inside the logarithm, which is a critical feature in graphing and analyzing these functions.

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