Download Lesson 13: Changing the Base

NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 13
M3
ALGEBRA II
Lesson 13: Changing the Base
Classwork
Exercises
1.
Assume that 𝑥, 𝑎, and 𝑏 are all positive real numbers, so that 𝑎 ≠ 1 and 𝑏 ≠ 1. What is log 𝑏 (𝑥) in terms of
log 𝑎 (𝑥)? The resulting equation allows us to change the base of a logarithm from 𝑎 to 𝑏.
2.
Approximate each of the following logarithms to four4 decimal places. Use the LOG key on your calculator rather
than logarithm tables, first changing the base of the logarithm to 10 if necessary.
3.
a.
log(32 )
b.
log 3 (32 )
c.
log 2 (32 )
In Lesson 12, we justified a number of properties of base 10 logarithms. Working in pairs, justify the following
properties of base 𝑏 logarithms.
a.
log 𝑏 (1) = 0
b.
log 𝑏 (𝑏) = 1
Lesson 13:
Date:
Changing the Base
9/17/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 13
M3
ALGEBRA II
4.
c.
log 𝑏 (b𝑟 ) = 𝑟
d.
𝑏 log𝑏 (𝑥) = 𝑥
e.
log 𝑏 (𝑥 ∙ 𝑦) = log 𝑏 (𝑥) + log 𝑏 (𝑦)
f.
log 𝑏 (𝑥 𝑟 ) = 𝑟 ∙ log 𝑏 (𝑥)
g.
log 𝑏 � � = −log 𝑏 (𝑥)
h.
log 𝑏 � � = log 𝑏 (𝑥) − log 𝑏 (𝑦)
1
𝑥
𝑥
𝑦
Find each of the following to 4four decimal places. Use the LN key on your calculator rather than a table.
a.
ln(32 )
b.
ln(24 )
Lesson 13:
Date:
Changing the Base
9/17/14
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Lesson 13
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
5.
6.
Write as a single logarithm:
1
a.
ln(4) − 3ln � � + ln(2)
b.
ln(5) + ln(32) − ln(4)
3
3
5
Write each expression as a sum or difference of constants and logarithms of simpler terms.
a.
ln �
b.
ln �
�5𝑥 3
𝑒2
�
(𝑥+𝑦)2
𝑥 2 +𝑦 2
�
Lesson 13:
Date:
Changing the Base
9/17/14
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Lesson 13
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Lesson Summary
We have established a formula for changing the base of logarithms from 𝑏 to 𝑎:
log 𝑏 (𝑥) =
log 𝑎 (𝑥)
.
log 𝑎 (𝑏)
In particular, the formula allows us to change logarithms base 𝑏 to common or natural logarithms, which are the
only two kinds of logarithms that calculators compute:
log b (𝑥) =
log(𝑥) ln(𝑥)
=
.
log(𝑏) ln(𝑏)
We have also established the following properties for base 𝑏 logarithms. If 𝑥, 𝑦, 𝑎 and 𝑏 are all positive real
numbers with 𝑎 ≠ 1 and 𝑏 ≠ 1 and 𝑟 is any real number, then:
1.
2.
3.
4.
5.
6.
7.
8.
log 𝑏 (1) = 0
log 𝑏 (𝑏) = 1
log 𝑏 (𝑏 𝑟 ) = 𝑟
𝑏 log𝑏 (𝑥) = 𝑥
log 𝑏 (𝑥 ∙ 𝑦) = log 𝑏 (𝑥) + log 𝑏 (𝑦)
log 𝑏 (𝑥 𝑟 ) = 𝑟 ∙ log 𝑏 (𝑥)
1
log 𝑏 � � = −log 𝑏 (𝑥)
𝑥
𝑥
log 𝑏 � � = log 𝑏 (𝑥) − log 𝑏 (𝑦)
𝑦
Problem Set
1.
Evaluate each of the following logarithmic expressions, approximating to four decimal places if necessary. Use the
LN or LOG key on your calculator rather than a table.
a.
b.
c.
2.
log 7 (11)
log 3 (2) + log 2 (3)
Use logarithmic properties and the fact that ln(2) ≈ 0.69 and ln(3) ≈ 1.10 to approximate the value of each of the
following logarithmic expressions. Do not use a calculator.
a.
b.
c.
d.
3.
log 8 (16)
ln(𝑒 4 )
ln(6)
ln(108)
8
ln � �
3
1
Compare the values of log 1 (10) and log 9 ( ) without using a calculator.
10
9
Lesson 13:
Date:
Changing the Base
9/17/14
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Lesson 13
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
4.
5.
6.
Show that for any positive numbers 𝑎 and 𝑏 with 𝑎 ≠ 1 and 𝑏 ≠ 1, log 𝑎 (𝑏) ∙ log 𝑏 (𝑎) = 1.
Express 𝑥 in terms of 𝑎, 𝑒, and 𝑦 if ln(𝑥) − ln(𝑦) = 2𝑎.
Re-write each expression in an equivalent form that only contains one base 10 logarithm.
a.
b.
c.
d.
7.
b.
c.
10
log 5 (12,500)
log 3 (0.81)
log 9 �√27�
log 8 (32)
1
log 4 � �
8
Write each expression as an equivalent expression with a single logarithm. Assume 𝑥, 𝑦, and 𝑧 are positive real
numbers.
a.
b.
c.
9.
1
log 𝑥 � �, for positive real values of 𝑥 not equal to 1
Write each number in terms of natural logarithms, and then use the properties of logarithms to show that it is a
rational number.
a.
8.
log 2 (800)
ln(𝑥) + 2 ln(𝑦) − 3 ln(𝑧)
1
2
(ln(𝑥 + 𝑦) − ln(𝑧))
(𝑥 + 𝑦) + ln(𝑧)
Rewrite each expression as sums and differences in terms of ln(𝑥), ln(𝑦), and ln(𝑧).
a.
b.
c.
ln(𝑥𝑦𝑧 3 )
ln �
𝑒3
𝑥𝑦𝑧
𝑥
�
ln �� �
𝑦
10. Solve the following equations in terms of base 5 logarithms. Then, use the change of base properties and a
th
calculator to estimate the solution to the nearest 1000 . If the equation has no solution, explain why.
P
a.
b.
c.
d.
5
2𝑥
= 20
75 = 10 ∙ 5𝑥−1
52+𝑥 − 5𝑥 = 10
2
5𝑥 = 0.25
11. In Lesson 6, you discovered that log(𝑥 ∙ 10𝑘 ) = 𝑘 + log(𝑥) by looking at a table of logarithms. Use the properties
of logarithms to justify this property for an arbitrary base 𝑏 > 0 with 𝑏 ≠ 1. That is, show that log 𝑏 (𝑥 ∙ 𝑏 𝑘 ) = 𝑘 +
log 𝑏 (𝑥).
Lesson 13:
Date:
Changing the Base
9/17/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 13
M3
ALGEBRA II
12. Larissa argued that since log 2 (2) = 1 and log 2 (4) = 2, then it must be true that log 2 (3) = 1.5. Is she correct?
Explain how you know.
13. EXTENSION. Suppose that there is some positive number 𝑏 so that:
log 𝑏 (2) = 0.36
log 𝑏 (3) = 0.57
a.
Use the given values of log 𝑏 (2), log 𝑏 (3), and log 𝑏 (5) to evaluate the following logarithms.
log b (6)
i.
log b (8)
log b (10)
ii.
iii.
log b (600)
iv.
b.
log 𝑏 (5) = 0.84.
Use the change of base formula to convert log 𝑏 (10) to base 10, and solve for 𝑏. Give your answer to 4
decimal places.
14. Solve the following exponential equations.
a.
b.
c.
d.
e.
23𝑥 = 16
2𝑥+3 = 43𝑥
34𝑥−2 = 27𝑥+2
1 3𝑥
42𝑥 = � �
5
0.2𝑥+3
4
= 625
15. Solve each exponential equation.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
32𝑥 = 81
63𝑥 = 36𝑥+1
625 = 53𝑥
254−𝑥 = 53𝑥
32𝑥−1 =
42𝑥
2𝑥−3
1
=1
82𝑥−4
𝑥
1
2
=1
2 = 81
8 = 3𝑥
6𝑥+2 = 12
10𝑥+4 = 27
2𝑥+1 = 31−𝑥
m. 32𝑥−3 = 2𝑥+4
Lesson 13:
Date:
Changing the Base
9/17/14
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 13
M3
ALGEBRA II
n.
o.
e2𝑥 = 5
𝑒 𝑥−1 = 6
16. In problem 9(e) of Lesson 12, you solved the equation 3𝑥 = 7−3𝑥+2 using the logarithm base 10.
a.
b.
c.
d.
Solve 3𝑥 = 7−3𝑥+2 using the logarithm base 3.
Apply the change of base formula to show that your answer to part (a) agrees with your answer to problem
9(e) of Lesson 12.
Solve 3𝑥 = 7−3𝑥+2 using the logarithm base 7.
Apply the change of base formula to show that your answer to part (c) also agrees with your answer to
problem 9(e) of Lesson 12.
17. Pearl solved the equation 2𝑥 = 10 as follows:
log(2𝑥 ) = log(10)
𝑥 log(2) = 1
1
.
𝑥=
log(2)
Jess solved the equation 2𝑥 = 10 as follows:
log 2 (2𝑥 ) = log 2 (10)
𝑥 log 2 (2) = log 2 (10)
𝑥 = log 2 (10).
Is Pearl correct? Is Jess correct? Explain how you know.
Lesson 13:
Date:
Changing the Base
9/17/14
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