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Transcript of Holt McDougal Algebra 2 7-5 Exponential and Logarithmic Equations and Inequalities 7-5 Exponential...
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities7-5 Exponential and Logarithmic Equations and Inequalities
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Algebra 2
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Opener-SAME SHEET-12/12
Solve for x
1.4x – 12 = 2x + 14
2. 3(x – 6) = 8 + 5x
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Example 2: Subtracting Logarithms
log749 – log77
log
27 + log13
13
19
c. log2 ( )5 1
2
c. 5log510
Evaluate log816.
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
7-4 Hmwk Quiz1. log
749 – log77
2. Evaluate log32
8.
3. log5252
4. log64 + log
69
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve exponential and logarithmic equations and equalities.
Solve problems involving exponential and logarithmic equations.
Objectives
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
• 7-5 Explore
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
exponential equationlogarithmic equation
Vocabulary
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Review Properties
Holt McDougal Algebra 2
7-5 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 2
7-5 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 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve and check.98 – x = 27x – 3
Example 1A: Solving Exponential Equations
x = 5
98 – x = 27x – 3
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve and check.4x – 1 = 5
Example 1B: Solving Exponential Equations
Check Use a calculator.
The solution is x ≈ 2.161.
x = 1 + ≈ 2.161log5log4
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve and check.
32x = 27
Check It Out! Example 1a
x = 1.5
7–x = 21
x = – ≈ –1.565log21log7
23x = 15
x ≈ 1.302
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Card Problems
Holt McDougal Algebra 2
7-5 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 2n > 106
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.
Write 1,000,000 in scientific annotation.
Take the log of both sides.log 2n > log 106
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Example 2 Continued
Use the Power of Logarithms.
log 106 is 6.nlog 2 > 6
nlog 2 > log 106
6log 2n > Divide both sides by log 2.
60.301n > 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.
Holt McDougal Algebra 2
7-5 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 3n – 1 > 1 x 108
$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.
Write 100,000,000 in scientific annotation.
Take the log of both sides.log 3n – 1 > log 108
Check It Out! Example 2
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Use the Power of Logarithms.
log 108 is 8.(n – 1)log 3 > 8
(n – 1) log 3 > log 108
8log 3n – 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
8log3n > + 1
Holt McDougal Algebra 2
7-5 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.
Raise to Same base
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Review the properties of logarithms from Lesson 7-4.
Remember!
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve.
Example 3A: Solving Logarithmic Equations
log6(2x – 1) = –1
7 12
x =
Solve.log
4100 – log4(x + 1) = 1
x = 24
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Opener-SAME SHEET-12/13Express each as a single logarithm.
1. log69 + log
624 log
6216 = 3
2. log3108 – log
34
Simplify.
3. log2810,000
log327 = 3
30,000
4. log44x –1 x – 1
5. 10log125 125
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve.
Example 3C: Solving Logarithmic Equations
log5x 4 = 8
x = 25
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve.Example 3D: Solving Logarithmic Equations
log12
x + log12
(x + 1) = 1
x(x + 1) = 12
log12
x + log12
(x +1) = 1x = 3 or x = –4
log12
x + log12
(x +1) = 1
log12
3 + log12
(3 + 1) 1log
123 + log
124 1
log12
12 1
The solution is x = 3.1 1
log12
( –4) + log12
(–4 +1) 1
log12
( –4) is undefined.
x
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Solve.
3 = log 8 + 3log x
Check It Out! Example 3a
5 = x
2log x – log 4 = 0
x = 2
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Watch out for calculated solutions that are not solutions of the original equation.
Caution
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Use a table and graph to solve 2x + 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 2
7-5 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 4aUse a table and graph to solve 2x = 4x – 1.
Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2.
The solution is x = 2.
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Team Problems
Solve.
1. 43x–1 = 8x+1
2. 32x–1 = 20
3. log7(5x + 3) = 3
4. log(3x + 1) – log 4 = 2
5. log4(x – 1) + log
4(3x – 1) = 2
x ≈ 1.86
x = 68
x = 133
x = 3
x = 5 3
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Opener-SAME SHEET-1/26In 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 billionThe 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?
4. Use a graph to predict the value in 4 years.
V(t)≈ 3000(0.7)t ≈ $720.30
Holt McDougal Algebra 2
7-5 Exponential and Logarithmic Equations and Inequalities
Review
• 9x = 3x-2 4x = 10 log6(2x + 3) = 3
Holt McDougal Algebra 2
7-5 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
Holt McDougal Algebra 2
7-5 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 4bUse a table and graph to solve 2x > 4x – 1.
Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2.
The solution is x < 2.
Holt McDougal Algebra 2
7-5 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 23x = 33x–1.
70 min
x ≈ 0.903