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

Solve for x 4x – 12 = 2x + 14 2. 3(x – 6) = 8 + 5x Opener-SAME SHEET-12/12 Solve for x 4x – 12 = 2x + 14 2. 3(x – 6) = 8 + 5x

Example 2: Subtracting Logarithms log749 – log77 c. log2 ( )5 1 2 log 27 + log 1 3 9 c. 5log510 Evaluate log816.

7-4 Hmwk Quiz log749 – log77 2. Evaluate log328. 3. log5252

Objectives Solve exponential and logarithmic equations and equalities. Solve problems involving exponential and logarithmic equations.

7-5 Explore

Vocabulary exponential equation logarithmic equation

Review Properties

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.

When you use a rounded number in a check, the result will not be exact, but it should be reasonable. Helpful Hint

Example 1A: Solving Exponential Equations 98 – x = 27x – 3 Solve and check. 98 – x = 27x – 3 x = 5

Example 1B: Solving Exponential Equations Solve and check. 4x – 1 = 5 x = 1 + ≈ 2.161 log5 log4 Check Use a calculator. The solution is x ≈ 2.161.

Check It Out! Example 1a Solve and check. 32x = 27 7–x = 21 23x = 15 log21 log7 x ≈ 1.302 x = 1.5

Card Problems

Example 2: Biology Application 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? At hour 0, there is one bacterium, or 20 bacteria. At hour one, there are two bacteria, or 21 bacteria, and so on. So, at hour n there will be 2n bacteria. Solve 2n > 106 Write 1,000,000 in scientific annotation. log 2n > log 106 Take the log of both sides.

Use the Power of Logarithms. Example 2 Continued nlog 2 > log 106 Use the Power of Logarithms. nlog 2 > 6 log 106 is 6. 6 log 2 n > Divide both sides by log 2. 6 0.301 n > Evaluate by using a calculator. n > ≈ 19.94 Round up to the next whole number. It will take about 20 hours for the number of bacteria to exceed 1,000,000.

Check It Out! Example 2 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. $1,000,000 is 100,000,000 cents. On day 1, you would receive 1 cent or 30 cents. On day 2, you would receive 3 cents or 31 cents, and so on. So, on day n you would receive 3n–1 cents. Solve 3n – 1 > 1 x 108 Write 100,000,000 in scientific annotation. log 3n – 1 > log 108 Take the log of both sides.

Check It Out! Example 2 Continued (n – 1) log 3 > log 108 Use the Power of Logarithms. (n – 1)log 3 > 8 log 108 is 8. 8 log 3 n – 1 > Divide both sides by log 3. 8 log3 n > + 1 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.

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. Raise to Same base

Review the properties of logarithms from Lesson 7-4. Remember!

Example 3A: Solving Logarithmic Equations Solve. Solve. log4100 – log4(x + 1) = 1 log6(2x – 1) = –1 7 12 x = x = 24

Express each as a single logarithm. 1. log69 + log624 log6216 = 3 Opener-SAME SHEET-12/13 Express each as a single logarithm. 1. log69 + log624 log6216 = 3 2. log3108 – log34 log327 = 3 Simplify. 3. log2810,000 30,000 4. log44x –1 x – 1 5. 10log125 125 22

Example 3C: Solving Logarithmic Equations Solve. log5x 4 = 8 x = 25

Example 3D: Solving Logarithmic Equations Solve. log12x + log12(x + 1) = 1 x(x + 1) = 12 x = 3 or x = –4 log12x + log12(x +1) = 1 log12x + log12(x +1) = 1 log123 + log12(3 + 1) 1 x log12( –4) + log12(–4 +1) 1 log123 + log124 1 log12( –4) is undefined. log1212 1 1 1  The solution is x = 3.

Check It Out! Example 3a Solve. 3 = log 8 + 3log x 2log x – log 4 = 0 x = 2 5 = x

Watch out for calculated solutions that are not solutions of the original equation. Caution

Example 4A: Using Tables and Graphs to Solve Exponential and Logarithmic Equations and Inequalities Use a table and graph to solve 2x + 1 > 8192x. 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}.

Check It Out! Example 4a Use a table and graph to solve 2x = 4x – 1. Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2. 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. The solution is x = 2.

Team Problems Solve. x = 5 3 1. 43x–1 = 8x+1 2. 32x–1 = 20 x ≈ 1.86 3. log7(5x + 3) = 3 x = 68 4. log(3x + 1) – log 4 = 2 x = 133 5. log4(x – 1) + log4(3x – 1) = 2 x = 3

2. Use a table to predict the population in 2020. Opener-SAME SHEET-1/26 In 2000, the world population was 6.08 billion and was increasing at a rate 1% each year. 1. Write a function for world population. Does the function represent growth or decay? P(t) = 6.08(1.01)t 2. Use a table to predict the population in 2020. ≈ 7.41 billion The value of a $3000 computer decreases about 30% each year. 3. Write a function for the computer’s value. Does the function represent growth or decay? V(t)≈ 3000(0.7)t ≈ $720.30 4. Use a graph to predict the value in 4 years. 30

Review 9x = 3x-2 4x = 10 log6(2x + 3) = 3

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

Check It Out! Example 4b Use a table and graph to solve 2x > 4x – 1. Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) 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 is x < 2.

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? 70 min 7. Use a table and graph to solve the equation 23x = 33x–1. x ≈ 0.903