Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve logarithmic equations. Objectives

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve and check. 4 x – 1 = 5 log 4 x – 1 = log 5 5 is not a power of 4, so take the log of both sides. (x – 1)log 4 = log 5 Apply the Power Property of Logarithms. Example 1 Divide both sides by log 4. Check Use a calculator. The solution is x ≈ x = 1 + ≈ log5 log4 x –1 = log5 log4

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve and check. 2 3x = 15 log2 3x = log15 15 is not a power of 2, so take the log of both sides. (3x)log 2 = log15 Apply the Power Property of Logarithms. Example 1b Divide both sides by log 2, then divide both sides by 3. x ≈ x = log15 log2

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities A logarithmic equation is an equation with a logarithmic expression that contains a variable. You can solve logarithmic equations by using the properties of logarithms.

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve. Example 2: Solving Logarithmic Equations log 6 (2x – 1) = –1

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve. Example 3: Solving Logarithmic Equations Write as a quotient. log – log 4 (x + 1) = x + 1 log 4 ( ) = 1

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve. Example 3B: Solving Logarithmic Equations Power Property of Logarithms. log 5 x 4 = 8 x = 25 Definition of a logarithm. 4log 5 x = 8 log 5 x = 2 x = 5 2 Divide both sides by 4 to isolate log 5 x.

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve. Example 3C: Solving Logarithmic Equations Product Property of Logarithms. log 12 x + log 12 (x + 1) = 1 log 12 x(x + 1) = 1

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Example 3 Continued Multiply and collect terms. Factor. Solve. x 2 + x – 12 = 0 log 12 x + log 12 (x +1) = 1 (x – 3)(x + 4) = 0 x – 3 = 0 or x + 4 = 0 Set each of the factors equal to zero. x = 3 or x = –4 log 12 x + log 12 (x +1) = 1 log log 12 (3 + 1) 1 log log log The solution is x = 3. 1 log 12 ( –4) + log 12 (–4 +1) 1 log 12 ( –4) is undefined. x Check Check both solutions in the original equation.

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve. 3 = log 8 + 3log x Check It Out! Example 3a 3 = log 8 + 3log x 3 = log 8 + log x 3 3 = log (8x 3 ) 10 3 = 10 log (8x 3 ) 1000 = 8x = x 3 5 = x Use 10 as the base for both sides. Use inverse properties on the right side. Product Property of Logarithms. Power Property of Logarithms.

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve. 2log x – log 4 = 0 Check It Out! Example 3b Write as a quotient. x = 2 Use 10 as the base for both sides. Use inverse properties on the left side. 2log ( ) = 0 x 4 2(10 log ) = 10 0 x 4 2( ) = 1 x 4

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Watch out for calculated solutions that are not solutions of the original equation. Caution

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities In the table, find the x-values where Y1 is equal to Y2. In the graph, find the x-value at the point of intersection. Check It Out! Example 4 Use a table and graph to solve 2 x = 4 x – 1. Use a graphing calculator. Enter 2 x as Y1 and 4 (x – 1) as Y2. The solution is x = 2.

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Lesson Quiz: Part I Solve x–1 = 8 x x–1 = log 7 (5x + 3) = 3 4. log(3x + 1) – log 4 = 2 5. log 4 (x – 1) + log 4 (3x – 1) = 2 x ≈ 1.86 x = 68 x = 133 x = 3 x = 5 3