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College Algebra K/DCMonday, 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.

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Evaluating Logarithms and the Change-of-Base Theorem4.4Common Logarithms ▪ Applications and Modeling with Common Logarithms

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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 0log534 2.7275 Logarithms commonly use 4

decimal places.

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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

3pH log H O , Memorize

Memorize

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Finding pH

Substitute.

Product property

log 10–8 = –8

83H 0 6.8 10 . Find the pH of a solution with

Then classify the solution.

The pH is 7.2, so the solution is alkaline (or basic).

Application: Complete-sentence answer.

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Find the hydronium ion concentration of a solution with pH = 4.3.

Finding pH

Substitute.

Multiply by –1.

Write in exponential form.

Use a calculator.

The hydronium concentration is 5.0 ͯ 10−5.

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Wetlands are classified as shown in the table.

Using pH in an Application

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.

≤

Copy this table for the assignment.

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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

0

10logI

dBI

Memorize

0

0

10,000,00010log

IdB

I

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Find the decibel rating of a sound with intensity 10,000,000I0.

Measuring the Loudness of Sound

The sound has a decibel rating of 70 dB.

Let I = 10,000,000I0.

log 10,000,000 = log 107 = 7.

Count the number of zeros.

010log

IdB

I

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Environmental Noise

Weakest sound heard ................................................................ 0 dBWhisper Quiet Library ……………………………………………… 30 dBNormal conversation (3-5') ………………………………………... 60 – 70 dBTelephone dial tone ……………………………………………….. 80 dBCity Traffic (inside car) …………………………………………….. 85 dBTrain whistle at 500', Truck Traffic ……………………………….. 90 dBSubway train at 200' ……………………………………………….. 95 dBLevel at which sustained exposuremay result in hearing loss …………………………………………. 90 – 95 dB Power mower at 3' …………………………………………………. 107 dBSnowmobile, Motorcycle …………………………………………... 100 dBPower saw at 3' …………………………………………………….. 110 dBSandblasting, Loud Rock Concert ……………………………….. 115 dBPain begins …………………………………………………………. 125 dBPneumatic riveter at 4' …………………………………………….. 125 dBEven short term exposure can causepermanent damage - Loudest recommendedexposure WITH hearing protection ………………………………. 140 dBJet engine at 100', Gun Blast ……………………………………... 140 dBDeath of hearing tissue ……………………………………………. 180 dBLoudest sound possible …………………………………………… 194 dB

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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.

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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 10–430) 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, 2.7 34) wine, 3.435) beer, 4.8 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 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

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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 100

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

100

log ,II

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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?

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College Algebra KTuesday, 08 April 2014

• OBJECTIVE TSW evaluate logarithms and use the change of base formula.

• ASSIGNMENTS DUE– 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 !!!

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Evaluating Logarithms and the Change-of-Base Theorem4.4Natural Logarithms ▪ Applications and Modeling with Natural Logarithms ▪ Logarithms with Other Bases

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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.

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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, log

loglog

.ba

b

xx

a

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Use the change-of-base theorem to find an approximation to four decimal places for each logarithm.

Change-of-Base Theorem

(a)

(b)

(a) (b)

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One measure of the diversity of the species in an ecological community is modeled by

Modeling Diversity of Species

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.)

You will need to copy this for the assignment.

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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.

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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. 453-454

(29-42 all, 45-50 all)

Separate assignments – do NOT combine on the same sheet of paper.

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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 0.88. Use the following values of n.

(a) 50 (b) 100 (c) 250

ln 1 .n

S n aa

Assignment: Sec. 4.4 - pp. 455-457 (53-56 all, 61-72 all)

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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.

Assignment: Sec. 4.4 - pp. 455-457 (53-56 all, 61-72 all)

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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.)

2 l6 ) og1 5 2 l6 ) og2 9 8 lo6 ) g3 0.59

8 lo6 ) g4 0.71 1 2 l6 og5) 3 1 3 l6 og6) 2

l6 ) og7 e l6 ) og8 2 13 lo6 g9) 12

19 l7 og0) 5 0.32 log71) 5 0.91 log72) 8