1 College Algebra K/DC Tuesday, 05 April 2016 OBJECTIVE TSW evaluate logarithms and use the change of base formula. TEST: Sec. 4.1 – 4.3 is not graded. TODAY’S ASSIGNMENTS (do not combine!) –Sec. 4.4: pp (63-68 all) Due tomorrow/Thursday, 04/06-07/16. –Sec. 4.4: p. 435 (29-42 all) Due on Friday, 04/08/16.

4-2 Due tomorrow/Thursday, 04/06-07/16. Assignment: pp (63-68 all) Due tomorrow/Thursday, 04/06-07/16. 63)Find the decibel rating of sounds having the following intensities. (a)100 I 0 (b)1000 I 0 (c)100,000 I 0 (d)1,000,000 I 0 (e)If the intensity of a sound is doubled, by how much is the decibel rating increased? 64)Find the decibel ratings of the following sounds, having intensities as given. Round each answer to the nearest whole number. (a)whisper, 115 I 0 (b)busy street, 9,500,000 I 0 (c)heavy truck, 20 m away, 1,200,000,000 I 0 (d)rock music, 895,000,000,000 I 0 (e)jetliner at takeoff, 109,000,000,000,000 I 0 65)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 I 0 is the amplitude of an earthquake of a certain (small) size. Find the Richter scale ratings for earthquakes having the following amplitudes. (a)1000I 0 (b)1,000,000I 0 (c)100,000,000I 0

4-3 Due tomorrow/Thursday, 04/06-07/16. Assignment: pp (63-68 all) Due tomorrow/Thursday, 04/06-07/16. 66)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 I 0. 67)On February 27, 2010, a massive earthquake struck Chile with a magnitude of 8.8 on the Richter scale. Express this reading in terms of I 0. 68)Compare your answers in Exercises 66 and 67. How many times greater was the force of the 2004 earthquake than that of the 2010 earthquake?

4-4 Due on Friday, 08 April Assignment: p. 435 (29-42 all) Due on Friday, 08 April 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 [H 3 O + ] for each substance with the given pH. 33)soda pop, 2.734)wine, )beer, 4.836)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

4-5 Evaluating Logarithms and the Change-of-Base Theorem 4.4 Common Logarithms ▪ Applications and Modeling with Common Logarithms

4-6 Common Logarithm For all positive numbers x, log x = log 10 x log x is the exponent to which 10 must be raised to get x. Ex: Logarithms commonly use 4 decimal places.

4-7 Measuring the Loudness of Sound The loudness of sound is measured in a unit called a d dd decibel. To measure loudness, we first assign an intensity of I 0 to a very faint sound, called the t tt threshold sound. If a particular sound has intensity I, then the decibel rating of this louder sound is Memorize

4-8 Find the decibel rating of a sound with intensity 10,000,000I 0. Measuring the Loudness of Sound The sound has a decibel rating of 70 dB. Let I = 10,000,000I 0. log 10,000,000 = log 10 7 = 7. Count the number of zeros.

4-9 Environmental Noise Weakest sound heard dB Whisper Quiet Library ………………………………………………30 dB Normal conversation (3-5') ………………………………………...60 – 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' ……………………………………………………..110 dB Sandblasting, Loud Rock Concert ………………………………..115 dB Pain begins ………………………………………………………….125 dB Pneumatic riveter at 4' ……………………………………………..125 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

Earthquakes and the Richter Scale The formula for calculating the intensity of an earthquake – as measured on the Richter Scale – is (a)What is the Richter Scale rating for an earthquake with amplitude I 0 ? 4-10 The Richter Scale rating is

Earthquakes and the Richter Scale The formula for calculating the intensity of an earthquake – as measured on the Richter Scale – is (b)How much more intense is this earthquake (8.696) than an earthquake with a rating of 6.432? 4-11 The higher-rated earthquake is times stronger than the lower-rated earthquake.

4-12 pH In chemistry, the pH of a solution is defined as where [H 3 O + ] is the hydronium ion concentration in moles per liter. pH measures the acidity or alkalinity of a solution. pH < 7.0acidic substances pH = 7.0pure water pH > 7.0alkaline (basic) substances Memorize

4-13 Finding pH Substitute. Product property log 10 –8 = –8 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.

4-14 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.

4-15 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.

4-16 Assignments Sec. 4.4: pp (63-68 all) –You do not have to write the problem, but you do have to show all work. –Due tomorrow/Thursday, 04/06-07/16. Sec. 4.4: p. 435 (29-42 all) –You do not have to write the problem, but you do have to show all work. –Due on Friday, 04/08/16.

17 College Algebra K/DC Wednesday, 06 April 2016 OBJECTIVE TSW evaluate logarithms and use the change of base formula. ASSIGNMENT DUE –Sec. 4.4: p (63-68 all)  I will collect ASSIGNMENT DUE FRIDAY, 04/08/16 –Sec. 4.4: p. 435 (29-42 all) ASSIGNMENT DUE BEFORE YOU LEAVE TODAY –Sec. 4.4: pp (71-74 all, all)

18 College Algebra K/DC Thursday, 07 April 2016 OBJECTIVE TSW evaluate logarithms and use the change of base formula. ASSIGNMENT DUE –Sec. 4.4: p (63-68 all)  I will collect ASSIGNMENT DUE TOMORROW, FRIDAY, 04/08/16 –Sec. 4.4: p. 435 (29-42 all) ASSIGNMENT DUE BEFORE YOU LEAVE TODAY –Sec. 4.4: pp (71-74 all, all)

4-19 Evaluating Logarithms and the Change-of-Base Theorem 4.4 Natural Logarithms ▪ Applications and Modeling with Natural Logarithms ▪ Logarithms with Other Bases

4-20 Natural Logarithms natural logarithm A logarithm with base e is a natural logarithm. ln x = log e x It is called a natural logarithm because it occurs in life sciences and economics in natural situations that involve growth and decay.

4-21 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,

4-22 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)

4-23 One measure of the diversity of the species in an ecological community is modeled by Modeling Diversity of Species where P 1, P 2, …, P n 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.

4-24 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

4-25 Assignment Sec. 4.4: pp (71-74 all, all)  black tray  black tray −71 – 74: Show work. 71, 72: Complete sentences not needed. Answers will be whole numbers. 73, 74: Complete sentence answers. (There are no units.) –79 – 90: 4 decimals Show work on all only.

)The number of species S(n) in a sample is given by where 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 72)In Exercise 71, find S(n) if a changes to Use the following values of n. (a)50(b)100(c)250 73)Suppose a sample of a small community shows two species with 50 individuals each. Find the measure of diversity H. 74)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. Due before you leave today  black tray Assignment: Sec pp (71-74 all, all) Due before you leave today  black tray

4-27 Due before you leave today  black tray Assignment: Sec pp (71-74 all, all) Due before you leave today  black tray 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.)