Exponential and Logarithmic Equations and Inequalities

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Exponential and Logarithmic Equations and
Inequalities
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Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Solve. 1. log16x 2. logx1.331
3 3. log10,000 x
64
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Objectives
Solve exponential and logarithmic equations and
equalities. Solve problems involving
exponential and logarithmic equations.
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Vocabulary
exponential equation logarithmic equation
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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.

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Check It Out! Example 1a
Solve and check.
32x 27
Rewrite each side with the same base 3 and 27
are powers of 3.
(3)2x (3)3
To raise a power to a power, multiply exponents.
32x 33
2x 3
Bases are the same, so the exponents must be
equal.
x 1.5
Solve for x.
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Check It Out! Example 1a Continued
Check
32x 27
32(1.5) 27
33 27
?
27 27
The solution is x 1.5.
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Check It Out! Example 1b
Solve and check.
7x 21
21 is not a power of 7, so take the log of both
sides.
log 7x log 21
Apply the Power Property of Logarithms.
(x)log 7 log 21
Divide both sides by log 7.
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Check It Out! Example 1b Continued
Check Use a calculator.
The solution is x 1.565.
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Check It Out! Example 1c
Solve and check.
23x 15
log23x log15
15 is not a power of 2, so take the log of both
sides.
Apply the Power Property of Logarithms.
(3x)log 2 log15
Divide both sides by log 2, then divide both
sides by 3.
x 1.302
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Check It Out! Example 1c Continued
Check Use a calculator.
The solution is x 1.302.
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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 3n1 cents.
Write 100,000,000 in scientific annotation.
Solve 3n 1 gt 1 x 108
log 3n 1 gt log 108
Take the log of both sides.
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Check It Out! Example 2 Continued
(n 1) log 3 gt log 108
Use the Power of Logarithms.
(n 1)log 3 gt 8
log 108 is 8.
Divide both sides by log 3.
Evaluate by using a calculator.
n gt 17.8
Round up to the next whole number.
Beginning on day 18, you would receive more than
a million dollars.
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Check It Out! Example 2
Check On day 18, you would receive 318 1 or
over a million dollars.
317 129,140,163 cents or 1.29 million dollars.
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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.
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Check It Out! Example 3a
Solve.
3 log 8 3log x
3 log 8 3log x
3 log 8 log x3
Power Property of Logarithms.
3 log (8x3)
Product Property of Logarithms.
103 10log (8x3)
Use 10 as the base for both sides.
1000 8x3
Use inverse properties on the right side.
125 x3
5 x
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Check It Out! Example 3b
Solve.
2log x log 4 0
Write as a quotient.
Use 10 as the base for both sides.
Use inverse properties on the left side.
x 2
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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 graph, find the x-value at the point of
intersection.
In the table, find the x-values where Y1 is equal
to Y2.
The solution is x 2.
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Check It Out! Example 4b
Use a table and graph to solve 2x gt 4x 1.
Use a graphing calculator. Enter 2x as Y1 and 4(x
1) as Y2.
In the graph, find the x-value at the point of
intersection.
In the table, find the x-values where Y1 is
greater than Y2.
The solution is x lt 2.
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Check It Out! Example 4c
Use a table and graph to solve log x2 6.
Use a graphing calculator. Enter log(x2) as Y1
and 6 as Y2.
In the graph, find the x-value at the point of
intersection.
In the table, find the x-values where Y1 is equal
to Y2.
The solution is x 1000.