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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