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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2:09 minutes

Problem 81

Textbook Question

In Exercises 79–82, use a graphing utility and the change-of-base property to graph each function.y = log2 (x + 2)

Verified step by step guidance

insert step 1> Identify the function to be graphed: \( y = \log_2(x + 2) \).

insert step 2> Use the change-of-base formula to rewrite the logarithmic function in terms of common logarithms: \( y = \frac{\log_{10}(x + 2)}{\log_{10}(2)} \).

insert step 3> Choose a range of x-values to evaluate the function, ensuring that \( x + 2 > 0 \) to keep the logarithm defined.

insert step 4> Use a graphing utility to plot the points obtained from the evaluated function and draw the curve.

insert step 5> Analyze the graph to understand its behavior, such as identifying the vertical asymptote at \( x = -2 \) and the domain \( x > -2 \).

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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 and are defined as y = log_b(x) if and only if b^y = x, where b is the base. They are used to solve for the exponent in equations involving powers. Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential for manipulating and graphing these functions.

Change-of-Base Formula

The change-of-base formula allows you to convert logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when using calculators or graphing utilities that may only support certain bases, such as base 10 or base e. This formula helps in evaluating logarithmic expressions and graphing them accurately.

Graphing Utilities

Graphing utilities, such as graphing calculators or software, are tools that allow users to visualize mathematical functions. They can plot functions, including logarithmic ones, by calculating values over a specified range. Understanding how to input functions correctly and interpret the resulting graphs is crucial for analyzing the behavior of logarithmic functions, such as their domain, range, and asymptotic behavior.