1. College Algebra K/DC Monday, 07 April 2014
OBJECTIVE TSW evaluate logarithms and use the change of base formula.
TESTS are not graded.
NEXT TEST (Sec. 4.4 – 4.6)
Wednesday, 16 April 2014.

2. Evaluating Logarithms and the Change-of-Base Theorem
4.4
Evaluating Logarithms and the Change-of-Base Theorem
Common Logarithms ▪ Applications and Modeling with Common Logarithms

3. 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:
Logarithms commonly use 4 decimal places.

4. pH In chemistry, the pH of a solution is defined as
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
Memorize

5. Finding pH Find the pH of a solution with 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).

6. 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.

7. 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.

8. 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
Memorize

9. Measuring the Loudness of Sound
Find the decibel rating of a sound with intensity 10,000,000I0.
Let I = 10,000,000I0.
Count the number of zeros.
log 10,000,000 = log 107 = 7.
The sound has a decibel rating of 70 dB.

10. Environmental Noise
Weakest sound heard dB
Whisper Quiet Library ……………………………………………… 30 dB
Normal conversation (3-5') ……………………………………… – 70 dB
Telephone dial tone ……………………………………………….. 80 dB
City Traffic (inside car) …………………………………………….. 85 dB
Train whistle at 500', Truck Traffic ……………………………….. 90 dB
Subway train at 200' ……………………………………………….. 95 dB
Level at which sustained exposure
may result in hearing loss …………………………………………. 90 – 95 dB
Power mower at 3' …………………………………………………. 107 dB
Snowmobile, Motorcycle ………………………………………… dB
Power saw at 3' …………………………………………………… dB
Sandblasting, Loud Rock Concert ……………………………… dB
Pain begins …………………………………………………………. 125 dB
Pneumatic riveter at 4' …………………………………………… dB
Even short term exposure can cause
permanent damage - Loudest recommended
exposure WITH hearing protection ………………………………. 140 dB
Jet engine at 100', Gun Blast …………………………………… dB
Death of hearing tissue ……………………………………………. 180 dB
Loudest sound possible …………………………………………… 194 dB

11. 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.

12. Assignment: pp. 453-454 (29-42 all, 45-50 all) Due on Wednesday, 09 April 2014.
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 30) limes, 1.6 x 10–2
31) crackers, 3.9 x 10–9 32) sodium hydroxide (lye),
3.2 x 10–14
Find the [H3O+] for each substance with the given pH.
33) soda pop, ) wine, 3.4
35) beer, ) 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 38) 6.22 x 10–5
39) 2.49 x 10–2 40) 3.14 x 10–2
41) 2.49 x 10–7 42) 5.86 x 10–7

13. 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
where I is the amplitude registered on a seismograph 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.
(a) 1000I0 (b) 1,000,000I0 (c) 100,000,000I0

14. 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?

15. College Algebra K Tuesday, 08 April 2014
OBJECTIVE TSW evaluate logarithms and use the change of base formula.
ASSIGNMENTS DUE
Sec. 4.2: pp (71-80 all, 82 omit d, 83, 84)  wire basket
Sec. 4.3: p. 442 (13-30 all)  black tray
Sec. 4.3: pp (59-69 odd, even, all)  to the right of the black tray
ASSIGNMENTS DUE TOMORROW
Sec. 4.4: p (29-42 all, all)
Sec. 4.4: pp (53-56 all, all)
Separate !!!

16. Evaluating Logarithms and the Change-of-Base Theorem
4.4
Evaluating Logarithms and the Change-of-Base Theorem
Natural Logarithms ▪ Applications and Modeling with Natural Logarithms ▪ Logarithms with Other Bases

17. 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.

18. 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,

19. 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)

20. 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.)

21. 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 = 200 members in the community, so and
Change-of-base theorem
The measure of diversity is

22. Assignment Sec. 4.4: pp. 455-457 (53-56 all, 61-72 all)
Due tomorrow, Wednesday, 09 April 2014.
ALSO DUE TOMORROW
Sec. 4.4: p (29-42 all, all)
Separate assignments – do NOT combine on the same sheet of paper.

23. Assignment: Sec. 4.4 - pp. 455-457 (53-56 all, 61-72 all)
53) The number of species in a sample is given by
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 Use the following values of n.
(a) 50 (b) 100 (c) 250

24. 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, Find the measure of diversity H.

25. 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 all.)