Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities 4-5 Exponential and Logarithmic Equations and Inequalities Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Algebra 2

4-5 Exponential and Logarithmic Equations and Inequalities Solve exponential and logarithmic equations and equalities. Solve problems involving exponential and logarithmic equations. Objectives

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities An exponential equation is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations: Try writing them so that the bases are all the same. Take the logarithm of both sides.

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities When you use a rounded number in a check, the result will not be exact, but it should be reasonable. Helpful Hint

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve and check. 9 8 – x = 27 x – 3 (3 2 ) 8 – x = (3 3 ) x – 3 Rewrite each side with the same base; 9 and 27 are powers of – 2x = 3 3x – 9 To raise a power to a power, multiply exponents. Example 1A: Solving Exponential Equations 16 – 2x = 3x – 9 Bases are the same, so the exponents must be equal. x = 5 Solve for x.

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 1B: Solving Exponential Equations 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. 3 2x = 27 (3) 2x = (3) 3 Rewrite each side with the same base; 3 and 27 are powers of x = 3 3 To raise a power to a power, multiply exponents. Check It Out! Example 1a 2x = 3 Bases are the same, so the exponents must be equal. x = 1.5 Solve for x.

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve and check. 7 –x = 21 log 7 –x = log is not a power of 7, so take the log of both sides. (–x)log 7 = log 21 Apply the Power Property of Logarithms. Check It Out! Example 1b Divide both sides by log 7. x = – ≈ –1.565 log21 log7 – x = log21 log7

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. Check It Out! Example 1c 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 Solve. Example 3A: Solving Logarithmic Equations Use 6 as the base for both sides. log 6 (2x – 1) = –1 6 log 6 (2x –1) = 6 –1 2x – 1 = x =x = Use inverse properties to remove 6 to the log base 6. Simplify.

Holt McDougal Algebra Exponential and Logarithmic Equations and Inequalities Solve. Example 3B: Solving Logarithmic Equations Write as a quotient. log – log 4 (x + 1) = 1 x = 24 Use 4 as the base for both sides. Use inverse properties on the left side. 100 x + 1 log 4 ( ) = 1 4 log 4 = x + 1 ( ) = x + 1

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

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 Use a table and graph to solve 2 x + 1 > 8192x. Example 4A: Using Tables and Graphs to Solve Exponential and Logarithmic Equations and Inequalities Use a graphing calculator. Enter 2^(x + 1) as Y1 and 8192x as Y2. In the table, find the x-values where Y1 is greater than Y2. In the graph, find the x-value at the point of intersection. The solution set is {x | x > 16}.

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