Logarithms Chapter 5
Logarithms Chapter 5. Inverse Functions An American traveling to Europe may find it confusing to find it only being 30 degree weather when they were told.
Dec 29, 2015
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Logarithms
Chapter 5
Inverse Functions
• An American traveling to Europe may find it confusing to find it only being 30 degree weather when they were told to pack shorts and bathing suits.
• If the function g converts Celsius to Fahrenheit temperatures, then the “inverse of g” would convert Fahrenheit back to Celsius.
Inverses• The inverse of g:
– Switches outputs to inputs instead of inputs to outputs
– Inverse of g ( ) undoes g
• Note: The inverse of a function is not necessarily a function
• If the inverse of a function is a function it is called invertible.
1g0)32(
32)0(1
g
g
Evaluating Inverse Functions
x f(x)
0 4
1 16
2 64
3 256
4 1024
5 4096
f(4)
)64(1f
Warning in , -1 is not an exponent, it is part of function notation to signify an inverse function
)(1 xf
Using Graphs
)5(
)5(
1f
f
Function vs Inverse (Tables)
x f(x)
45 .01
90 .1
135 1
180 10
225 100
270 1000
x f(x)
.01 45
.1 90
1 135
10 180
100 225
1000 270
Graphs
1f
is the reflection of f across the line x=y for any invertible function f
1f
How to plot an inverse function
• Plot f
• Chose points on f
• For each point (a,b), plot points (b,a)
• Sketch the inverse
Example
(-7,-1)→(-1,-7)
(4,2)→(2,4)
Finding an Equation of the Inverse
• find– substitute s for
1f
415)( txf
)(xf
26.06.15
4
15
1
15
15
15
4
44154
415
st
st
ts
ts
ts
26.06.)(1 txf
One-to-one
• Function - each input has exactly one output
• One-to-one function - each output originates from exactly one input– All one-to-one functions are invertible– Linear functions with nonzero slopes are one-
to-one functions/invertible
5.2 Logarithmic Functions
Logarithm
• Inverse of an exponential function
• For b > 0, b≠ 1, and a > 0,– the logarithm is the number k such that
)(log)(
)(1 xxf
bxf
b
x
4
813
81log3
x
xx
ablog.abk
ab
kak
b
log
x
b
bxg
xxg
)(
)(log)(1
Logarithm Properties
kk
kn
nn
kn
/1
33/13/1
388
1log
8828
3
18log
3
12log
nn
nn
1
1
5
1log
55
15log
1
01log
19
01log
0
0
9
n
n
ab
ka
k
b
1
1log
64
14
364
1log
3
4
nn
nn
2/1
21
4
2
1log
44
2
14log
kk
kn
nn
kn
b
b
b
log
5
5
6log66
65
Common Logarithm
• A logarithm with a base of 10
• 4 decimal places right
• 5 decimal places left
• Only works with 1
4)10000log(
aa 10loglog
5)00001log(.
...60.44000010)40000log( x
Logarithmic Function
• Base, b, is a function that can be put in the form
• where b > 0 and b ≠ 1xxf 5)(
)(log)( xxf b
125
5)(
)3(3
xf
f
4
)625(log)(
)625(
51
1
xf
f
Graphing x y
1 0
2 1
4 2
8 3
x y
0 1
1 2
2 4
3 8
)(log)( 2 xxf
xxf 2)(
)(log)( 2 xxf
xxf 2)(
Logarithmic Models
• In Aug 2011, an earthquake in Virginia had an amplitude of times the reference amplitude . In Jan 2011, California had an earthquake with an amplitude of
times the reference amplitude .• Find the Richter number of the
earthquakes.• Find the ratio of the amplitudes.
5103.6
4103.1 0A
0A
Logarithmic Functions (Cont)
8.5
103.6log
103.6log
5
0
05
A
AR
1.4
103.1log
103.1log
4
0
04
A
AR
5.48103.1
103.6
04
05
A
A
Virginia earthquake was 48.5 times greater than the California earthquake
5.3 Solving Equations of Logarithms/Exponents
4
64
64
3)64(log
3/13/13
3
b
b
b
b
243
3
5)(log5
3
x
x
x
16
4096
4096
4
6)(log
3 33
3
36
34
x
x
x
x
x
abka kb log
abka kb log
574
4
4
228
4228
1515415243
154243
1543
5154log5
3
x
x
x
x
x
x
x
16
2
4)(log3
12
3
)(log3
12)(log3
41644)(log3
164)(log3
4
2
2
2
2
2
x
x
x
x
x
x
x
Properties of Logarithms
• Power Property– For x > 0, b > 0 and b ≠ 1
• Equality Property– For positive real numbers a, b, c, b ≠ 1
)(loglog xpx bp
b
)(log)(log caandca bb
)(log2)(log 32
3 xx
288)(
8
8loglog
33/13/13
3
43
4
x
x
x
Solving Exponential Equations
3)3log(
)27log(
)3log(
)27log(
)3log(
)3log(
)27log()3log(
)27log(3log
273
x
x
x
x
x
sparenthesiremember
)3log()27log(3log(27log(
33
9
3
3
93
63663
363
)4log(
)64log(
)4log(
)4log()63(
)64log()4log()63(
)64log()4log(
64463
63
x
x
x
x
x
x
x
x
x
Solving Using Graphing
xx 52
)2.1,2(.
)6.22,5.4(
5.4 Making Predictions with Exponential Models
Change-of-Base Property• For a > 0, b > 0, a ≠ 1, b ≠ 1, and x > 0.
• Since you can change a to any number, using base 10 makes it easy to plug into your calculator for computing logarithms not in base 10
)(log
)(log)(log
b
xx
a
ab
)4log(
)6log(
)4(log
)6(log
)4(log
)6(log)6(log
7
7
2
24
Example 1
• A person invests $500,000 in an account at 6.5% annual interest, after winning the lottery. Let V=f(t) be the value in dollars of the account after t years.
• Write an equation
ttfV )065.1(000,500)(
065.11065.100
5.6
• What is the V-intercept?– 500,000
• What does this represent?– This is the initial deposit
• What is the rate of growth?– 1.065 – 1 = .065*100 = 6.5
• What does this represent?– The value is growing by 6.5% per year.
• Find f(5)
• What does this represent?– The value of the account will be $685,000
after 5 years
000,685
)370.1(500000
)065.1(500000)( 5
tf
• Find f(t) = 1,000,000
• What does this represent?– The value will reach $1,000,000 (double) after 11
years
11
)065.1log(
)2log(
)2(log
)065.1(2
000,500
)065.1(000,500
000,500
000,000,1
)065.1(000,500000,000,1
065.1
t
t
t
t
t
t
Example 2• World Population in billions is given in the
table below. Year Population
1930 (0) 2.070
1940 (10) 2.295
1950 (20) 2.500
1960 (30) 3.050
1970 (40) 3.700
1980 (50) 4.454
1990 (60) 5.279
2000 (70) 6.080
• Graph a scatterplot of the data
• Pick two points on the curve– (10, 2.295) (60, 5.279)
• Find an equation
017.1
)()3.2(
3.2
295.2
279.5
)(295.2
279.5
50/15050/1
50
1060
10
60
b
b
b
b
divideab
ab
ttfP
a
a
a
a
)017.1(938.1)(
938.1184.1
184.1
184.1
295.2
184.1295.2
)017.1(295.2 10
Graph the Line to Check It
• What is the rate of growth?– 1.017 – 1 = .017*100 = 1.7
• What does this represent?– The populations is growing by 1.7% per year
• What is the P-intercept?– 1.938
• What does this represent?– That the world population (at t = 0) in 1930
was approximately 1.938 billion
• Predict when the world population will reach 10 billion.
• 1930 + 97 = 2027Approximately the year 2027
97
)017.1log(
)160.5log(
)160.5(log
)017.1(160.5
938.1
)017.1(938.1
938.1
10
)017.1(938.110
017.1
t
t
t
t
t
t
Example 3
• Suppose a virus is spreading among a population at an average rate of 2.5% of the population per day. If there are currently 506 people already infected on Oct. 10th, 2011. On what day will the amount of infected people be doubled t days after Oct. 10th.
• Write a formula.
• Find when the infected population will double.
28 days after Oct. 10th ~ Nov. 7th is when the population of infected people will be double
ttI )025.1(506)(
025.11025.100
5.2
506
)025.1(506
506
1012
)025.1(5061012
1012)506(2
t
t
28
)025.1log(
)2log(
)2(log
)025.1(2
025.1
t
t
t
t
• When will the number of people infected be tripled.
– About 44 days after Oct 10th ~ Nov 23rd is when the population of infected people will be tripled
44
)025.1log(
)3log(
)3(log
)025.1(3
506
)025.1(506
506
1518
)025.1(5061518
1518)3(506
025.1
t
t
t
t
t
t
• What is the I-intercept?– 506
• What does this number represent?– The number of infected people (at t = 0) on
Oct. 10th, 2011
• What is the rate of growth?– 1.025 – 1 = .025*100 = 2.5
• What does it represent for this situation?– The percent at which the virus is spreading
through the population
• In 2005, a crater was found in a desolate area thought to be formed by a collapsed volcano. If the amount of carbon-14 present in a charcoal sample can be used to determine when the crater formed and the charcoal had 97.32% of the carbon-14 remaining, estimate how long ago it formed. The half-life of carbon-14 is 5730 years.
Example 4
• We know the amount at t = 0 is 100%– S-intercept is (0, 100)
• We know at 5730 years, the amount will be at 50%.– (5730, 50)
• Write an equation
5730
5730
5730
5.
100
100
100
50
10050
100)(
b
b
b
btS t
ttS
b
b
)99988(.100)(
99988.
)()5(. 5730/157305730/1
• When was the crater formed?
The crater is 226 years old. 2005 – 226 = 1779 The crater was formed in 1779.
226
)99988log(.
)9732log(.
)9732(.log
9732.)99988(.
100
32.97
100
)99988(.100
32.97)99988(.100
99988.
t
t
t
t
t
t
5.5 More Logarithm Properties
Product Property
• For x > 0, y > 0, b > 0, and b ≠ 1,
– The sum of logarithms is the logarithm of the product of their inputs.
)(log)(log)(log xyyx bbb
)9(log)(log)9(log.1 22
222 xyyx
)8(log)8(log)8(log)(log
)2(log)(log)2(log3)(log4.2
63
6123
63
123
323
433
23
33
xxxxx
xxxx
Quotient Property
• For x > 0, y > 0, b > 0, and b ≠ 1,
– The difference of two logarithms is the logarithm of the quotient of their inputs
y
xyx bbb log)(log)(log
36
4
664
6 2log2
4log)2(log)4(log x
x
xxx
Examples
7
8
212
105
212105
122105
122105
34252
42
27
8log
108
32log
)108log()32log(
)274log()32log(
)27log()4log()32log(
)3log()2log()2log(
)3log(3)2log(2)2log(5
x
y
yx
yx
yxyx
xyyx
xyyx
xyxy
xyxy
22
9log
216
72log
2)16(log)72(log
2)16(log)98(log
2)16(log)9(log)8(log
2)2(log)3(log)2(log
2)2(log4)3(log2)2(log3
512
12
7
12
1212
712
1212
4312
1212
412
312
4312
2212
312
312
21212
x
x
x
xx
xxx
xxx
xxx
xxx
2
1
32
1
32
1)(
32
1
288
9
9288
22
9)2)(144(
2
912
22
9log
5
5/15/15
5
5
55
5
52
512
x
x
x
xx
x
x
x
Graphing Logarithmic Functions
)4log(
)2log()2(log4
xx
Solving with Graphing
53)8log(
)log(
53)(log8
xx
xx
(1.76, .27)
x ≈ 1.76
Warning
)(log
)(log)(log
b
xx
a
ab
y
xyx bbb log)(log)(log
)(log
)(loglog
)(log
)(log)(log)(log
b
x
y
x
b
xyx
a
ab
a
abb
5.6 Natural Logarithms
Natural Logarithms
• Logarithm with base e– e ≈ 2.71828182…
• Note: e is a constant, irrational number, NOT a variable
)(log)ln( aa e
aeca c )ln(
5
)(log
)ln(
5
5
5
x
ee
xe
e
x
e
7914.754,162
12)ln(12
x
xe
x
5444.3653
3
3
3
3
7)3ln(
3433)3ln(
43)3ln(2
8
2
3)3ln(2
83)3ln(2
7
7
7
x
ex
xe
xe
x
x
x
x
x
7081.4
2)15ln(
2)15ln(
15
6
90
6
6
906
2
2
2
x
x
x
e
e
e
x
x
x
4091.16
4)86ln(6
6
6
4)86ln(
64)86ln(
4464)86ln(
86
86
86
46
1254
1254
x
x
x
x
x
e
e
ee
x
xx
xx
Properties of Natural Logarithms
)ln()ln(
ln)ln()ln(
)ln()ln()ln(
)ln()ln(
1)ln(
0)1ln(
yxyx
y
xyx
xyyx
xpx
ep
194
3
57
3
)4(3
57)4(3
35433)4(3
543)4(3
x
x
x
x
x
1240.2
)4ln(
)19ln(
)4ln(
)19ln(
)4ln(
)4ln(
)19ln()4ln(
)19ln()4ln(
x
x
x
x
x
9ln
9ln
9ln
)9ln()ln(
)3ln()ln(
)3ln(2)ln(9
12
618
6
18
618
2392
32
x
x
x
x
xx
xx
xx
125
16ln)
125
16ln(
125
16ln
)125ln()16ln(
)125ln()16ln(
)125ln()ln()16ln(
)5ln()ln()2ln(
)5ln(3)ln(5)2ln(4
26632
6
32
632
62012
62012
325443
243
xx
x
x
xx
xxx
xxx
xxx
xxx
5)64ln(
5)64ln(
581
5184ln
5)81ln()5184ln(
5)81ln()8164ln(
5)81ln()81ln()64ln(
5)9ln()3ln()2ln(
5)9ln(2)3ln(4)2ln(6
8
614
6
14
614
686
686
23426
32
x
x
x
x
xx
xxx
xxx
xxx
xxx
1109.1
64
)(64
64
64
64
64
5)64ln(
8/15
8/18
8/15
85
85
8
x
ex
xe
xe
xe
x
Logarithmic Models
A person makes chicken soup. The temperature of the soup decreases by the equation:
Minutes Temp (F)
0 200
1 194
2 187
3 182
4 176
5 171
tetH 05.013268)(
Graph a scattergram to check the equation
• What was the temperature of the soup when it was made (t=0)?– 200ºF (y-intercept)
• If a person waits 6 minutes for the soup to cool before eating, what will the temperature be?
– The soup is approximately 166ºF
7856.165)6(
7856.9768
)7408(.13268
13268
132683.
)6(05.0
h
e
e
• The soup will be “lukewarm” when it reaches a temperature of 98.6ºF, how long will it take to become “lukewarm”?
– The soup will be lukewarm after approximately 29 minutes
t
t
t
t
e
e
e
e
05.0
05.0
05.0
05.0
2318.
132
132
132
6.30
1326868686.98
132686.98
2376.2905.0
)2318ln(.05.0
05.0
05.0
)2318ln(.
t
t
t
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