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Holt Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities 7-5 Exponential and Logarithmic Equations and Inequalities Holt Algebra 2 Warm.

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Presentation on theme: "Holt Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities 7-5 Exponential and Logarithmic Equations and Inequalities Holt Algebra 2 Warm."— Presentation transcript:

1 Holt Algebra Exponential and Logarithmic Equations and Inequalities 7-5 Exponential and Logarithmic Equations and Inequalities Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

2 Holt Algebra Exponential and Logarithmic Equations and Inequalities Warm Up Solve. 1. log 16 x = 2. log x = 3 3. log10,000 = x

3 Holt Algebra Exponential and Logarithmic Equations and Inequalities Solve exponential and logarithmic equations and equalities. Solve problems involving exponential and logarithmic equations. Objectives

4 Holt Algebra Exponential and Logarithmic Equations and Inequalities exponential equation logarithmic equation Vocabulary

5 Holt 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.

6 Holt 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

7 Holt 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.

8 Holt Algebra Exponential and Logarithmic Equations and Inequalities Example 1A Continued Check 9 8 – x = 27 x – – – The solution is x = 5.

9 Holt 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

10 Holt 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.

11 Holt Algebra Exponential and Logarithmic Equations and Inequalities Check 3 2x = (1.5) The solution is x = 1.5. Check It Out! Example 1a Continued

12 Holt 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

13 Holt Algebra Exponential and Logarithmic Equations and Inequalities Check Use a calculator. The solution is x ≈ – Check It Out! Example 1b Continued

14 Holt 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

15 Holt Algebra Exponential and Logarithmic Equations and Inequalities Check Use a calculator. The solution is x ≈ Check It Out! Example 1c Continued

16 Holt Algebra Exponential and Logarithmic Equations and Inequalities Suppose a bacteria culture doubles in size every hour. How many hours will it take for the number of bacteria to exceed 1,000,000? Example 2: Biology Application Solve 2 n > 10 6 At hour 0, there is one bacterium, or 2 0 bacteria. At hour one, there are two bacteria, or 2 1 bacteria, and so on. So, at hour n there will be 2 n bacteria. Write 1,000,000 in scientific annotation. Take the log of both sides. log 2 n > log 10 6

17 Holt Algebra Exponential and Logarithmic Equations and Inequalities Example 2 Continued Use the Power of Logarithms. log 10 6 is 6. nlog 2 > 6 nlog 2 > log log 2 n > Divide both sides by log n > Evaluate by using a calculator. n > ≈ Round up to the next whole number. It will take about 20 hours for the number of bacteria to exceed 1,000,000.

18 Holt Algebra Exponential and Logarithmic Equations and Inequalities Example 2 Continued Check In 20 hours, there will be 2 20 bacteria = 1,048,576 bacteria.

19 Holt Algebra Exponential and Logarithmic Equations and Inequalities You receive one penny on the first day, and then triple that (3 cents) on the second day, and so on for a month. On what day would you receive a least a million dollars. Solve 3 n – 1 > 1 x 10 8 $1,000,000 is 100,000,000 cents. On day 1, you would receive 1 cent or 3 0 cents. On day 2, you would receive 3 cents or 3 1 cents, and so on. So, on day n you would receive 3 n–1 cents. Write 100,000,000 in scientific annotation. Take the log of both sides. log 3 n – 1 > log 10 8 Check It Out! Example 2

20 Holt Algebra Exponential and Logarithmic Equations and Inequalities Use the Power of Logarithms. log 10 8 is 8. (n – 1)log 3 > 8 (n – 1) log 3 > log log 3 n – 1 > Divide both sides by log 3. Evaluate by using a calculator. n > ≈ 17.8 Round up to the next whole number. Beginning on day 18, you would receive more than a million dollars. Check It Out! Example 2 Continued 8 log3 n > + 1

21 Holt Algebra Exponential and Logarithmic Equations and Inequalities Check On day 18, you would receive 3 18 – 1 or over a million dollars = 129,140,163 cents or 1.29 million dollars. Check It Out! Example 2

22 Holt 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.

23 Holt Algebra Exponential and Logarithmic Equations and Inequalities Review the properties of logarithms from Lesson 7-4. Remember!

24 Holt 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.

25 Holt 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

26 Holt 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.

27 Holt 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

28 Holt 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.

29 Holt 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.

30 Holt 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

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

32 Holt 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}.

33 Holt Algebra Exponential and Logarithmic Equations and Inequalities log(x + 70) = 2log ( ) In the table, find the x-values where Y1 equals Y2. In the graph, find the x-value at the point of intersection. x 3 Use a graphing calculator. Enter log(x + 70) as Y1 and 2log( ) as Y2. x 3 The solution is x = 30. Example 4B

34 Holt 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 4a 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.

35 Holt Algebra Exponential and Logarithmic Equations and Inequalities 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. Check It Out! Example 4b 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.

36 Holt 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 4c Use a table and graph to solve log x 2 = 6. Use a graphing calculator. Enter log(x 2 ) as Y1 and 6 as Y2. The solution is x = 1000.

37 Holt 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

38 Holt Algebra Exponential and Logarithmic Equations and Inequalities Lesson Quiz: Part II 6. A single cell divides every 5 minutes. How long will it take for one cell to become more than 10,000 cells? 7. Use a table and graph to solve the equation 2 3x = 3 3x–1. 70 min x ≈ 0.903


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