Transcript Evaluating Logarithms and the Change of Base Formula

College Algebra K/DC
Monday, 07 April 2014
•
OBJECTIVE TSW evaluate logarithms and use
•
TESTS are not graded.
•
NEXT TEST (Sec. 4.4 – 4.6)
the change of base formula.
–
Wednesday, 16 April 2014.
1
4.4
Evaluating Logarithms and the
Change-of-Base Theorem
Common Logarithms ▪ Applications and Modeling with Common
Logarithms
4-2
Common Logarithm
For all positive numbers x,
log x = log10x
log x is the exponent to which 10 must be raised
to get x.
Ex:
log100  2
log10  1
log1  0
log534  2.7275
Logarithms commonly use 4
decimal places.
4-3
pH
In chemistry, the pH of a solution is defined as
pH   log H3O  , Memorize


where [H3O+] is the hydronium ion concentration in
moles per liter.
pH measures the acidity or alkalinity of a solution.
pH < 7.0 acidic substances
pH = 7.0 pure water
pH > 7.0 alkaline (basic) substances
Memorize
4-4
Finding pH

8

H
0

6.8

10
.
Find the pH of a solution with  3 
Then classify the solution.
Substitute.
Product property
log 10–8 = –8
Application: Complete-sentence answer.
The pH is 7.2, so the solution is alkaline (or basic).
4-5
Finding pH
Find the hydronium ion concentration of a solution
with pH = 4.3.
Substitute.
Multiply by –1.
Write in
exponential form.
Use a calculator.
The hydronium concentration is 5.0 ͯ 10−5.
4-6
Using pH in an Application
Wetlands are
classified as shown in
the table.
≤
Copy this table for
the assignment.
The hydronium ion concentration of a water sample
from a wetland is 4.5 x 10–3. Classify this wetland.
The wetland is a bog because the pH ≤ 3.0.
4-7
Measuring the Loudness of Sound
The loudness of sound is measured in a unit called
a decibel.
To measure loudness, we first assign an intensity
of I0 to a very faint sound, called the threshold
sound.
If a particular sound has intensity I, then the
decibel rating of this louder sound is
I
dB  10log
Memorize
I0
4-8
Measuring the Loudness of Sound
Find the decibel rating of a sound with intensity
10,000,000I0.
I
dB  10log
I0
10,000,000I0
Let I = 10,000,000I0.
dB  10log
I0
Count the number of zeros.
log 10,000,000
= log 107 = 7.
The sound has a decibel rating of 70 dB.
4-9
Environmental Noise
Weakest sound heard ................................................................
Whisper Quiet Library ………………………………………………
Normal conversation (3-5') ………………………………………...
Telephone dial tone ………………………………………………..
City Traffic (inside car) ……………………………………………..
Train whistle at 500', Truck Traffic ………………………………..
Subway train at 200' ………………………………………………..
Level at which sustained exposure
may result in hearing loss ………………………………………….
Power mower at 3' ………………………………………………….
Snowmobile, Motorcycle …………………………………………...
Power saw at 3' ……………………………………………………..
Sandblasting, Loud Rock Concert ………………………………..
Pain begins ………………………………………………………….
Pneumatic riveter at 4' ……………………………………………..
Even short term exposure can cause
permanent damage - Loudest recommended
exposure WITH hearing protection ……………………………….
Jet engine at 100', Gun Blast ……………………………………...
Death of hearing tissue …………………………………………….
Loudest sound possible ……………………………………………
0 dB
30 dB
60 – 70 dB
80 dB
85 dB
90 dB
95 dB
90 – 95 dB
107 dB
100 dB
110 dB
115 dB
125 dB
125 dB
140 dB
140 dB
180 dB
194 dB
4-10
Assignment
• Sec. 4.4: p. 453-454 (29-42 all, 45-50 all)
– You do not have to write the problem, but you do have
to show all work.
– Due on Wednesday, 09 April, 2014.
4-11
Assignment: pp. 453-454 (29-42 all, 45-50 all)
Wednesday, 09 April 2014.
Due on
For each substance, find the pH from the given hydronium ion
concentration. Then classify as acidic or alkaline (basic).
29) grapefruit, 6.3 x 10–4
31) crackers, 3.9 x 10–9
30) limes, 1.6 x 10–2
32) sodium hydroxide (lye),
3.2 x 10–14
Find the [H3O+] for each substance with the given pH.
33) soda pop, 2.7
35) beer, 4.8
34) wine, 3.4
36) drinking water, 6.5
Suppose that water from a wetland area is sampled and found to have the
given hydronium ion concentration. Determine whether the wetland is a
rich fen, a poor fen, or a bog.
37) 2.49 x 10–5
39) 2.49 x 10–2
41) 2.49 x 10–7
38) 6.22 x 10–5
40) 3.14 x 10–2
42) 5.86 x 10–7
4-12
Assignment: p. 454 (45-50 all)
45) Find the decibel rating of sounds having the following intensities.
(a) 100 I0
(b) 1000 I0
(c) 100,000 I0
(d) 1,000,000 I0
(e) If the intensity of a sound is doubled, by how much is the
decibel rating increased?
46) Find the decibel ratings of the following sounds, having intensities as
given. Round each answer to the nearest whole number.
(a) whisper, 115 I0
(b) busy street, 9,500,000 I0
(c) heavy truck, 20 m away, 1,200,000,000 I0
(d) rock music, 895,000,000,000 I0
(e) jetliner at takeoff, 109,000,000,000,000 I0
47) The magnitude of an earthquake, measured on the Richter scale, is
I
log10 , where I is the amplitude registered on a seismograph 100
I0
km from the epicenter of the earthquake, and I0 is the amplitude of an
earthquake of a certain (small) size. Find the Richter scale ratings for
earthquakes having the following amplitudes.
4-13
(a) 1000I
(b) 1,000,000I (c)
100,000,000I
Assignment: p. 454 (45-50 all)
48) On December 26, 2004, the third largest earthquake ever recorded
struck in the Indian Ocean with a magnitude of 9.1 on the Richter scale.
The resulting tsunami killed an estimated 229,900 people in several
countries. Express this reading in terms of I0.
49) On March 28, 2005, the seventh largest earthquake ever recorded struck
in Northern Sumatra, Indonesia, with a magnitude of 8.6 on the Richter
scale. Express this reading in terms of I0.
50) Compare your answers in Exercises 48 and 49. How many times greater
was the fource of the 2004 earthquake than the 2005 earthquake?
4-14
College Algebra K
Tuesday, 08 April 2014
•
OBJECTIVE TSW evaluate logarithms and use the
•
ASSIGNMENTS DUE
change of base formula.
–
–
–
•
–
–
Sec. 4.2: pp. 429-430 (71-80 all, 82 omit d, 83, 84)
wire basket

Sec. 4.3: p. 442 (13-30 all)  black tray
Sec. 4.3: pp. 443-444 (59-69 odd, 70-80 even, 81-88
all)  to the right of the black tray
ASSIGNMENTS DUE TOMORROW
Sec. 4.4: p. 453-454 (29-42 all, 45-50 all)
Sec. 4.4: pp. 455-457 (53-56 all, 61-72 all)
Separate !!!
15
4.4
Evaluating Logarithms and the
Change-of-Base Theorem
Natural Logarithms ▪ Applications and Modeling with Natural
Logarithms ▪ Logarithms with Other Bases
4-16
Natural Logarithms
A logarithm with base e is a natural logarithm.
ln x = loge x
It is called a natural logarithm because it occurs in
life sciences and economics in natural situations
that involve growth and decay.
4-17
Logarithms with Other Bases
You can use a calculator to find the values of
either common logarithms (base 10) or natural
logarithms (base e).
For logarithms of other bases, you must use the
“changed of base” formula.
Change-of-Base Theorem
For any positive real numbers x, a, and b, where
a ≠ 1 and b ≠ 1,
logb x
loga x 
.
logb a
4-18
Change-of-Base Theorem
Use the change-of-base theorem to find an approximation
to four decimal places for each logarithm.
(a)
(b)
(a)
(b)
4-19
Modeling Diversity of Species
One measure of the diversity of the species in an
ecological community is modeled by
You will need to copy this for the assignment.
where P1, P2, …, Pn are the proportions of a sample
that belong to each of n species found in the sample.
(Source: Ludwig, J., and J. Reynolds, Statistical Ecology: A Primer on
Methods and Computing, New York, Wiley, 1988, p. 92.)
4-20
Modeling Diversity of Species
Find the measure of diversity in a community with two
species where there are 60 of one species and 140 of
the other.
There are 60 + 140 = 200 members in the community,
so
and
Change-of-base
theorem
The measure of
diversity is 0.881.
4-21
Assignment
• Sec. 4.4: pp. 455-457
(53-56 all, 61-72 all)
− Due tomorrow,
Wednesday, 09 April 2014.
Separate assignments – do NOT
combine on the same sheet of
paper.
• ALSO DUE TOMORROW
− Sec. 4.4: p. 453-454
(29-42 all, 45-50 all)
4-22
Assignment: Sec. 4.4 - pp. 455-457 (53-56 all, 61-72 all)
53) The number of species in a sample is given by
 n
S  n   a ln  1   .
 a
Here n is the number of individuals in the sample, and a is
a constant that indicates the diversity of species in the
community. If a = 0.36, find S(n) for each value of n. (Hint:
S(n) must be a whole number.)
(a) 100
(b) 200
(c) 150
(d) 10
54) In Exercise 53, find S(n) if a changes to 0.88. Use the
following values of n.
(a) 50
(b) 100
(c) 250
4-23
Assignment: Sec. 4.4 - pp. 455-457 (53-56 all, 61-72 all)
55) Suppose a sample of a small community shows two
species with 50 individuals each. Find the measure of
diversity H.
56) A virgin forest in northwestern Pennsylvania has 4 species
of large trees with the following proportions of each:
hemlock, 0.521; beech, 0.324; birch, 0.081; maple, 0.074.
Find the measure of diversity H.
4-24
Assignment: Sec. 4.4 - pp. 455-457 (53-56 all, 61-72 all)
Use the change-of-base theorem to find an approximation to four decimal
places for each logarithm. (Write the problem and solve. Show work on 61-64
all.)
61) log 2 5
62) log 2 9
63) log 8 0.59
64) log 8 0.71
65) log 1 2 3
66) log 1 3 2
67) log  e
68) log  2
69) log
71) log 0.32 5
72) log 0.91 8
70) log
19
5
13
12
4-25