College Algebra Chapter 4 Exponential and Logarithmic Functions Section 4.4 Properties of Logarithms Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Apply the Product, Quotient, and Power Properties of Logarithms Write a Logarithmic Expression in Expanded Form Write a Logarithmic Expression as a Single Logarithm Apply the Change-of-Base Formula

Concept 1 Apply the Product, Quotient, and Power Properties of Logarithms

Apply the Product, Quotient, and Power Properties of Logarithms Let b, x, and y be positive real numbers where b ≠ 1. Product Property : Quotient Property : Power Property : For these exercises, assume that all variable expressions represent positive real numbers.

Examples 1 – 3 Use the product property of logarithms to write the logarithm as a sum. Then simplify if possible. log (2xy) = log 2 + log x + log y ln (3(a + b)) = ln 3 + ln (a + b)

Skill Practice 1 Write the logarithm as a sum and simplify if possible. Assume that a, c, and d represented positive real numbers.

Examples 4 – 6 Use the quotient property of logarithms to write the logarithm as a difference. Then simplify if possible.

Skill Practice 2 Write the logarithm as the difference of logarithms and simplify if possible. Assume that t represents a positive real number.

Examples 7 – 9 Apply the power property of logarithms.

Skill Practice 3 Apply the power property of logarithms.

Concept 2 Write a Logarithmic Expression in Expanded Form

Example 10 Write the expression as the sum or difference of logarithms. Solution:

Example 11 Write the expression as the sum or difference of logarithms. Solution:

Example 12 Write the expression as the sum or difference of logarithms. Solution:

Example 13 Write the expression as the sum or difference of logarithms. Solution:

Example 14 Write the expression as the sum or difference of logarithms. Solution:

Example 15 Write the expression as the sum or difference of logarithms. Solution:

Skill Practice 4 Write the expression as the sum or difference of logarithms.

Concept 3 Write a Logarithmic Expression as a Single Logarithm

Example 16 Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. log (4x – 3) – log x Solution:

Example 17 Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. Solution:

Example 18 Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. 3 ln x + ln (x - 2) – ln 5 Solution:

Example 19 Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. Solution:

Example 20 Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. Solution:

Skill Practice 5 Write the expression as a single logarithm and simplify the result if possible.

Skill Practice 6 Write the expression as a single logarithms and simplify the results if possible.

Examples 21 – 23

Skill Practice 7

Concept 4 Apply the Change-of-Base Formula

Apply the Change-of-Base Formula Let a and b be positive real numbers such that a ≠ 1 and b ≠ 1. Then for any positive real number x In particular,

Examples 24, 25 Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places.

Skill Practice 8 Estimate between two consecutive integers. Use the change-of-base formula to evaluate by using base 10. Round to 4 decimal places. Use the change-of-base formula by using base e. Round to 4 decimal places. Check the result by using the related exponential form.