In Exercises 79–82, use a graphing utility and the change-of-base property to graph each function. y = log3 x

Verified step by step guidance

1. <Step 1: Understand the function.> The function given is \( y = \log_3 x \), which is a logarithmic function with base 3. This means that for any value of \( x \), \( y \) is the power to which 3 must be raised to get \( x \).
2. <Step 2: Use the change-of-base formula.> The change-of-base formula allows us to rewrite the logarithm in terms of a different base, typically base 10 or base \( e \) (natural logarithm). The formula is \( \log_b a = \frac{\log_c a}{\log_c b} \). For this problem, we can rewrite \( \log_3 x \) as \( \frac{\log_{10} x}{\log_{10} 3} \) or \( \frac{\ln x}{\ln 3} \).
3. <Step 3: Graph the function using a graphing utility.> Input the rewritten function into a graphing calculator or software. For example, if using base 10, enter \( y = \frac{\log_{10} x}{\log_{10} 3} \). If using natural logarithms, enter \( y = \frac{\ln x}{\ln 3} \).
4. <Step 4: Analyze the graph.> Observe the shape and behavior of the graph. The graph of \( y = \log_3 x \) should pass through the point (1,0) because \( \log_3 1 = 0 \). It should also approach negative infinity as \( x \) approaches 0 from the right, and increase without bound as \( x \) increases.
5. <Step 5: Identify key features.> Note the vertical asymptote at \( x = 0 \) and the fact that the function is defined only for \( x > 0 \). The graph should be increasing and concave down.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The function y = log_b(x) answers the question, 'To what power must the base b be raised to obtain x?' Understanding the properties of logarithms, such as their domain, range, and behavior, is essential for graphing and analyzing these functions.

Change-of-Base Formula

The change-of-base formula allows you to convert logarithms from one base to another, which is particularly useful when using calculators that only compute logarithms in base 10 or base e. The formula is expressed as log_b(x) = log_k(x) / log_k(b), where k is any positive number. This concept is crucial for graphing logarithmic functions with different bases.

Graphing Utilities

Graphing utilities, such as graphing calculators or software, enable users to visualize mathematical functions. They can plot functions, including logarithmic ones, and help in understanding their behavior, such as intercepts, asymptotes, and overall shape. Familiarity with these tools enhances the ability to analyze and interpret the graphs of functions effectively.