Download Resource Notes - NUAMES Mathematics

ge.cii.o.M 5-5,5-6, ~
5-7
Sections 5-5, 3-6, and 5-7 focus on logarithms: evaluating and applying
logarithms (including natural logarithms), and using laws of logarithms and
the change-of-base formula. Section 5-7 also presents a technique for solving
certain types of real-world problems by using logarithms to solve
exponential equations.
KEYTE~MS
EXflMPLE/lLLOSTItfITlOI't
Common logarithm of a positive number x (p. 191)
the exponent you get when x is written as a power of 10
(10, 1)
log /0 = log 10-1 =-1
log 1 = log 10° = 0
log 10 = log 101 = 1
Logarithm of a positive number x to the base b where b > 0 and
b -;f. 1 (p. 193)
the exponent you get when x is written as a power of b
Natural logarithm of a positive number x (p. 193)
(written In x or log e x) the exponent you get when
a power of e
Base b logarithmic function with b > 0 and b
the inverse of the base b exponential function
-;f.
x
=
10g28
is written as
3, since 23
=
8
In 4.0552 "" 1.4, since
e 1.4 "" 4.0552
1 (p. 193)
Natural logarithmic function (p. 193)
the inverse of the natural exponential function, f(x)
.
Y
,'!(x)=eX,."
3 (1, e)/
,.,.)
= eX
I
2
I
,
I
,.
<~'"
,.
,.
,.
3
x
Exponential equation (p. 203)
an equation that contains a variable in the exponent
52
ADVANCED MATHEMATICS
Student Resource Guide
Copyright © by Houghton Mifflin Company. All rights reserved.
O"DERSTAHDIHG THE MAl" IDEAS
Logarithms
• A logarithm is an exponent.
x = b" if and only if log , x = a
x = ek if and only ifln x = k
• Laws of Logarithms
M, N, and b are positive real numbers and b
1. log , MN
2. log ,
M
N
=
=
::f:-
1.
log , M + logj, N
•
log , M - 10gb N
3. log , M = log , N if and only if M = N
4. 10gb Mk
=
5. change-of-base formula: 10gb c
=
log
5
•
log42"
•
If log x
k log , M, for any real number k c
a b -oga
-1
=
log412
In x5
=
=
log42
+ log46
log, 5 - log, 2
=
=
log 8, then x
8
51n x
log 8
In 8
= ~
log 3
In 3
log 8 = -3
Exponential equations
=
4x-l
32
(22)x-l = 25
22x-2 = 25
• If possible, write each side of the equation as a power of
the same number. Then set the exponents equal.
2x - 2 = 5
x = 3.5
• If you cannot write each side of the equation as a power
of the same number, take the logarithm of each side
and apply law 4 listed above.
---_.
4x-1
log 4x-1
=
x-I
=
log 30
log 4
=
1 + l~
30
log 30
(x - 1) log 4 = log 30
=
x
~o "" 3.45
CHECKI"G THE MAl" IDEAS
For Exercises 1-4, match each logarithm with a logarithm equal
to it. Do not use a calculator.
A. log 4 - 3
1. log34
2. log 64
B. 3 log 4
3. log 12
log 4
C. log 3
4. log 0.004
D. log 4 + log 3
5. Write In 20 "" 3 in exponential form.
6. Find each logarithm. (Do not use a calculator; see Example 2 on text
page 193.)
a. log , 81
1
b. log2 16
Copyright © by Houghton Mifflin Company. All rights reserved.
d. logs 1
ADVANCED MATHEMATICS
Student Resource Guide 53
7. Express
(~)2
in terms of log M and log N. (See Examples 1 and 2
lO;N
on text page 198.)
8. Simplify ~(log5 50 + log , 12.5). (See Example 3 on text page 198.)
9. Critical Thinking Use your calculator to evaluate each of the
expressions log 108.2, 10 log 8.2, In e8.2, and eln 8.2. Then write two
general laws suggested by these expressions and explain why each must
be true.
10. Solve (1.7)X
11. Solve 25x
= 18.
-[5
= --. x
12S
(lSlrtG THE MAlrt IDEAS
Example 1 Given log 4 "" 0.6021, find:
a. log 40
b. log 0.25
mmm
c. log 2
Write each logarithm in term of log 4.
a. log 40 = log(4 • 10) = log 4 + log 10 "" 0.6021 + 1
b. log 0.25
c. log 2
=
=
log
log,f4
i=
=
log 4 -1
log 4112
=
=
1.6021
-l(log 4) "" -0.6021
= ~ log 4
"" ~(0.6021) "" 0.3011
Example Z Simplify 102 + 4 log x .
mmm
102+4logx
=
=
=
=
102 • 104logx
100 • 101ogx4
100 • x4
f-bX+Y
ff-
= b+»
bY
log Mk = k • log M
l O" and log x are inverse functions.
100x4
Exercises
12. Given log 9 "" 0.9542, find:
a. log
1
81
c. log 27
b. log 900
13. Writing Suppose you use your calculator to approximate log 20 and
your calculator displays 2.30103. Write a convincing argument to
explain why this value must be incorrect.
14. Simplify (a) e 1 + In 2 and (b) 100.5 log
15. Express y in terms of x if log y
=
0
9.
2 log x - 1.
(See Example 4 on text page 198.)
16. Application If you invest $100 at 6% annual
interest compounded monthly, when will the
investment be worth $150? (See Example 2 on
text page 204).
54
ADVANCED MATHEMATICS
Student Resource Guide
Copyright © by Houghton Mifflin Company. All rights reserved.