Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Properties of Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Properties of Logarithms

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1:44 minutes

Problem 79

Textbook Question

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

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 \( x \).

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 \( \frac{\ln x}{\ln 3} \).

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 \( y = \frac{\ln x}{\ln 3} \).

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.

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.