College Algebra Chapter 4 Exponential and Logarithmic Functions Section 4.3 Logarithmic Functions Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Convert Between Logarithmic and Exponential Forms Evaluate Logarithmic Expressions Apply Basic Properties of Logarithms Graph Logarithmic Functions Use Logarithmic Functions in Applications

Concept 1 Convert Between Logarithmic and Exponential Forms

Convert Between Logarithmic and Exponential Forms Logarithmic Function: If x and b are positive real numbers such that b ≠ 1, then is called the logarithmic function with base b where

Examples 1 – 4 Write each equation in exponential form.

Skill Practice 1 Write each equation in exponential form.

Examples 5 – 7 Write each equation in logarithmic form.

Skill Practice 2 Write each equation in logarithmic form.

Concept 2 Evaluate Logarithmic Expressions

Examples 8 – 10 Evaluate each logarithmic expression.

Skill Practice 3 Evaluate each expression.

Evaluate Logarithmic Expressions Common logarithmic function: Natural logarithmic function:

Examples 11 – 14 Evaluate each expression.

Skill Practice 4 Evaluate. log 10,000,000 log 0.1 ln e

Examples 15 – 18 Use your calculator to find the approximate value. Round the answer to 4 decimal places. Check your answer by using the exponential form. Example: log 43 ≈ 1.6335

Skill Practice 5 Approximate the logarithms. Round to 4 decimal places. log 229 ln 0.0216 ln 87 ln 0.0032

Concept 3 Apply Basic Properties of Logarithms

Apply Basic Properties of Logarithms

Examples 19 – 22 Simplify each expression.

Examples 23 – 27 Simplify each expression.

Skill Practice 6 Simplify.

Concept 4 Graph Logarithmic Functions

Example 28 (1 of 2) (rewrite into exponential form and select values for y first) Exponential form:

Example 28 (2 of 2)

Skill Practice 7 Graph the functions.

Example 29 (1 of 3) Graph Solution: x y = f(x) 2 1 3 ≈ 3.6 4 -1 -1/2

Example 29 (2 of 3)

Example 29 (3 of 3) Where in If a < 0 reflect across the x-axis. Shrink vertically if 0 < |a| < 1. Stretch vertically if |a| > 1. (x-h): If h > 0, shift to the right. If h < 0, shift to the left. k: If k > 0, shift upward. If k < 0, shift downward.

Graph Logarithmic Functions (1 of 2) For b > 1

Graph Logarithmic Functions (2 of 2) Domain: (-∞, ∞) Range: (0, ∞) Horizontal Asymptote: y = 0 Passes through: (0, 1) Domain: (0, ∞) Range: (-∞, ∞) Vertical Asymptote: x = 0 Passes through: (1, 0)

Skill Practice 8 Graph the function. Identify the vertical asymptote and write the domain in interval notation.

Example 30 (1 of 2) Give the domain and range in interval notation. Determine the vertical asymptote.

Example 30 (2 of 2) Solution: Domain: (-3, ∞) Range: (-∞, ∞) VA: x = -3

Example 31 (1 of 2) Give the domain and range in interval notation. Determine the vertical asymptote.

Example 31 (2 of 2) Solution: Domain: (0, ∞) Range: (-∞, ∞) VA: x = 0 (y axis)

Example 32 (1 of 2) Give the domain and range in interval notation. Determine the vertical asymptote.

Example 32 (2 of 2) Solution: VA: x = 4, x = -1 Boundary points: x = 4, x = -1

Skill Practice 9 Write the domain in interval notation.

Concept 5 Use Logarithmic Functions in Applications

Example 33 (1 of 3) The absolute magnitude, M, of a star is the apparent magnitude, m, a star would have if it were placed 10 parsecs from earth. The lower the value of the magnitude, the brighter the star. Our sun has an apparent magnitude of –26.74. The brightest star in our night sky is Sirius, the Dog Star, with an apparent magnitude of –1.44. The sun appears so bright because it is very close (astronomically speaking). The formula relates a star’s absolute magnitude, apparent magnitude, and its distance, d, from earth in parsecs.

Example 33 (2 of 3) If Sirius is 2.637 parsecs from earth and the sun is parsecs from earth, what is the absolute magnitude of each star? Solution: Sirius: m = -1.44 d = 2.637

Example 33 (3 of 3) Sun: m = -26.74

Skill Practice 10 Determine the magnitude of an earthquake that is How many times more intense is a 5.2-magnitude earthquake than a 4.2-magnitude earthquake?