Solved: Suppose log b2 = a and log b3 = c. Use the

Problem 3E Write the exponential equation in logarithmic form. 34 = 81

Problem 1E Write the exponential equation in logarithmic form. 53 = 125

Problem 2E Write the exponential equation in logarithmic form. 72 = 49

Problem 4E Write the exponential equation in logarithmic form. 27 = 128

Problem 94A Music Theory A music theorist associates the fundamental frequency of a pitch f with a real number defined by p = 69 + 12 log2 (f/440). Source: Science. a. Standard concert pitch for an A is 440 cycles per second. Find the associated value of p. ________________ b. An A one octave higher than standard concert pitch is 880 cycles per second. Find the associated value of p.

Problem 5E Write the exponential equation in logarithmic form.

Problem 6E Write the exponential equation in logarithmic form.

Problem 7E Write the logarithmic equation in exponential form. log2 32 = 5

Problem 8E Write the logarithmic equation in exponential form. log3 81 = 4

Problem 9E Write the logarithmic equation in exponential form.

Problem 10E Write the logarithmic equation in exponential form.

Problem 11E Write the logarithmic equation in exponential form. log 100,000 = 5

Problem 12E Write the logarithmic equation in exponential form. log 0.001 = ?3

Problem 13E Evaluate the logarithm without using a calculator. log8 64

Problem 14E Evaluate the logarithm without using a calculator. log9 81

Problem 15E Evaluate the logarithm without using a calculator. log4 64

Problem 16E Evaluate the logarithm without using a calculator. log3 27

Problem 18E Evaluate the logarithm without using a calculator.

Problem 17E Evaluate the logarithm without using a calculator.

Problem 20E Evaluate the logarithm without using a calculator.

Problem 21E Evaluate the logarithm without using a calculator. ln e

Problem 22E Evaluate the logarithm without using a calculator. ln e3

Problem 19E Evaluate the logarithm without using a calculator.

Problem 24E Evaluate the logarithm without using a calculator. ln 1

Problem 23E Evaluate the logarithm without using a calculator. ln e5/3

Problem 25E Is the “logarithm to the base 3 of 4” written as log4 3 or log3 4?

Problem 26E Write a few sentences describing the relationship between ex and ln x.

Problem 27E Use the properties of logarithms to write the expression as a sum, difference, or product of simpler logarithms. For example, log5(3k)

Problem 29E Use the properties of logarithms to write the expression as a sum, difference, or product of simpler logarithms. For example,

Problem 28E Use the properties of logarithms to write the expression as a sum, difference, or product of simpler logarithms. For example, log9(4m)

Problem 30E Use the properties of logarithms to write the expression as a sum, difference, or product of simpler logarithms. For example,

Problem 31E Use the properties of logarithms to write the expression as a sum, difference, or product of simpler logarithms. For example,

Problem 32E Use the properties of logarithms to write the expression as a sum, difference, or product of simpler logarithms. For example,

Problem 33E Suppose log b2 = a and log b3 = c. Use the properties of logarithms to find the following. log b 32

Problem 34E Suppose log b2 = a and log b3 = c. Use the properties of logarithms to find the following. log b 18

Problem 36E Suppose log b 2 = a and log b3 = c. Use the properties of logarithms to find the following. log b(9b2)

Problem 39E Use natural logarithms to evaluate the logarithm to the nearest thousandth. log1.2 0.95

Problem 38E Use natural logarithms to evaluate the logarithm to the nearest thousandth. log12 210

Problem 40E Use natural logarithms to evaluate the logarithm to the nearest thousandth. log2.8 0.12

Problem 41E Solve each equation. Round decimal answers to four decimal places. log x 36 = ?2

Problem 35E Suppose log b2 = a and log b3 = c. Use the properties of logarithms to find the following. log b (72b)

Problem 37E Use natural logarithms to evaluate the logarithm to the nearest thousandth. log5 30

Problem 42E Solve each equation. Round decimal answers to four decimal places. log9 27 = m

Problem 43E Solve each equation. Round decimal answers to four decimal places. log8 16 = z

Problem 46E Solve each equation. Round decimal answers to four decimal places. log4(5x + 1) = 2

Problem 48E Solve each equation. Round decimal answers to four decimal places. log4x ? log4(x + 3) = ?1

Problem 50E Solve the equation. Round decimal answers to four decimal places. log (x + 5) + log(x + 2) = 1

Problem 44E Solve each equation. Round decimal answers to four decimal places.

Problem 47E Solve each equation. Round decimal answers to four decimal places. log5(9x ? 4) = 1

Problem 51E Solve each equation. Round decimal answers to four decimal places. log3(x ? 2) + log3(x + 6) = 2

Problem 49E Solve each equation. Round decimal answers to four decimal places. log9m ? log9(m ? 4) = ?2

Problem 45E Solve each equation. Round decimal answers to four decimal places.

Problem 52E Solve each equation. Round decimal answers to four decimal places. log3(x2 + 17) ? log3(x + 5) = 1

Problem 53E Solve each equation. Round decimal answers to four decimal places. log2(x2 ? 1)? log2(x + 1) = 2

Problem 55E Solve equation in Exercise. Round decimal answers to four decimal places. ln x + ln 3x = ?1

Problem 56E Solve equation in Exercise. Round decimal answers to four decimal places. ln (x + 1) ? ln x = 1

Problem 54E Solve equation in Exercise. Round decimal answers to four decimal places. ln (5x + 4) = 2

Problem 57E Solve each equation. Round decimal answers to four decimal places. 2x = 6

Problem 58E Solve each equation. Round decimal answers to four decimal places. 5x = 12

Problem 61E Solve equation in Exercise. Round decimal answers to four decimal places. 3x+1 = 5x

Problem 59E Solve each equation. Round decimal answers to four decimal places. ek?1 = 6

Problem 60E Solve each equation. Round decimal answers to four decimal places. e2y = 15

Problem 62E Solve equation in Exercise. Round decimal answers to four decimal places. 2x+1 = 6x-1

Problem 63E Solve each equation. Round decimal answers to four decimal places. 5(0.10)x = 4(0.12)x

Problem 64E Solve each equation. Round decimal answers to four decimal places. 1.5(1.05)x = 2(1.01)x

Problem 65E Write expression using base e rather than base 10. 10x+1

Problem 66E Write expression using base e rather than base 10.

Problem 70E Find the domain of the function. f(x) = ln(x2 ? 9)

Problem 67E Approximate expression in the form ax without using e. e3x

Problem 68E Approximate expression in the form ax without using e. e?4x

Problem 71E Lucky Larry was faced with solving log(2x + 1) ? log(3x ? 1) = 0. Larry just dropped the logs and proceeded: Although Lucky Larry is wrong in dropping the logs, his procedure will always give the correct answer to an equation of the form log A ? log B = 0, where A and B are any two expressions in x. Prove that this last equation leads to the equation A? B = 0, which is what you get when you drop the logs. Source: The AMATYC Review.

Problem 72E Find all errors in the following calculation.

Problem 75A Inflation Assuming annual compounding, find the time it would take for the general level of prices in the economy to double at the following annual inflation rates. a. 3% b. 6% c. 8% d. Check your answers using either the rule of 70 or the rule of 72, whichever applies.

Problem 73E Prove:

Problem 74E Prove: loga xr = r loga x.

Problem 69E Find the domain of the function. f(x) = log(5 ? x)

Problem 76A Interest Mary Klingman invests $15,000 in an account paying 7% per year compounded annually. a. How many years are required for the compound amount to at least double? (Note that interest is only paid at the end of each year.) ________________ b. In how many years will the amount at least triple? ________________ c. Check your answer to part a using either the rule of 70 or the rule of 72, whichever applies.

Problem 77A Interest Leigh Jacks plans to invest $500 into a money market account. Find the interest rate that is needed for the money to grow to $1200 in 14 years if the interest is compounded continuously. (Compare with Exercise 1.) Exercise 1 Interest Leigh Jacks plans to invest $500 into a money market account. Find the interest rate that is needed for the money to grow to $1200 in 14 years if the interest is compounded quarterly.

Problem 78A Rule of 72 Complete the following table, and use the results to discuss when the rule of 70 gives a better approximation for the doubling time, and when the rule of 72 gives a better approximation. r 0.001 0.02 0.05 0.08 0.12 (ln 2)/ln(1 + r) 70/100r 72/100r

Problem 79A Pay Increases You are offered two jobs starting July 1, 2013. Humongous Enterprises offers you $45,000 a year to start, with a raise of 4% every July 1. At Crabapple Inc. you start at $30,000, with an annual increase of 6% every July 1. On July 1 of what year would the job at Crabapple Inc. pay more than the job at Humongous Enterprises? Use the algebra of logarithms to solve this problem, and support your answer by using a graphing calculator to see where the two salary functions intersect.

Problem 83A Index of Diversity, refer to Example. Index of Diversity One measure of the diversity of the species in an ecological community is given by the index of diversityH, where and P1, P2, . . . , Pn are the proportions of a sample belonging to each of n species found in the sample. Source: Statistical Ecology. For example, in a community with two species, where there are 90 of one species and 10 of the other, P1 = 90/100 = 0.9 and P2 = 10/100 = 0.1, with Verify that H ? 0.673 if there are 60 of one species and 40 of the other. As the proportions of nspecies get closer to 1/n each, the index of diversity increases to a maximum of ln n. Find the value of the index of diversity for populations with n species and 1/n of each if a.n = 3; ________________ b.n = 4. ________________ c. Verify that your answers for parts a and b equal ln 3 and ln 4, respectively.

Problem 81A Index of Diversity, refer to Example. Index of Diversity One measure of the diversity of the species in an ecological community is given by the index of diversityH, where and P1, P2, . . . , Pn are the proportions of a sample belonging to each of n species found in the sample. Source: Statistical Ecology. For example, in a community with two species, where there are 90 of one species and 10 of the other, P1 = 90/100 = 0.9 and P2 = 10/100 = 0.1, with Verify that H ? 0.673 if there are 60 of one species and 40 of the other. As the proportions of nspecies get closer to 1/n each, the index of diversity increases to a maximum of ln n. Suppose a sample of a small community shows two species with 50 individuals each. a. Find the index of diversity H. ________________ b. What is the maximum value of the index of diversity for two species? ________________ c. Does your answer for part a equal ln 2? Explain why.

Problem 82A Index of Diversity, refer to Example. Index of Diversity One measure of the diversity of the species in an ecological community is given by the index of diversityH, where and P1, P2, . . . , Pn are the proportions of a sample belonging to each of n species found in the sample. Source: Statistical Ecology. For example, in a community with two species, where there are 90 of one species and 10 of the other, P1 = 90/100 = 0.9 and P2 = 10/100 = 0.1, with Verify that H ? 0.673 if there are 60 of one species and 40 of the other. As the proportions of nspecies get closer to 1/n each, the index of diversity increases to a maximum of ln n. 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 index of diversity H.

Problem 84A Allometric Growth The allometric formula is used to describe a wide variety of growth patterns. It says that y = nxm, where x and y are variables, and n and m are constants. For example, the famous biologist J. S. Huxley used this formula to relate the weight of the large claw of the fiddler crab to the weight of the body without the claw. Source: Problems of Relative Growth. Show that if x and y are given by the allometric formula, then X = log bx, Y = log by, and N = log bn are related by the linear equation Y = mX + N.

Problem 80A Insect Species An article in Science stated that the number of insect species of a given mass is proportional to m?0.6, where m is the mass in grams. Source: Science. A graph accompanying the article shows the common logarithm of the mass on the horizontal axis and the common logarithm of the number of species on the vertical axis. Explain why the graph is a straight line. What is the slope of the line?

Problem 87A Minority Population The U.S. Census Bureau has reported that the United States is becoming more diverse. In Exercise of the previous section, the projected Hispanic population (in millions) was modeled by the exponential function h(t) = 37.79 (1.021)t where t = 0 corresponds to 2000 and 0 ?t ? 50. Source: U.S. Census Bureau. a. Estimate in what year the Hispanic population will double the 2005 population of 42.69 million. Use the algebra of logarithms to solve this problem. ________________ b. The projected U.S. Asian population (in millions) was modeled by the exponential function h (t) = 11.14 (1.023)t, where t = 0 corresponds to 2000 and 0 ?t ? 50. Estimate in what year the Asian population will double the 2005 population of 12.69 million. Minority Population According to the U.S. Census Bureau, the United States is becoming more diverse. Based on U.S. Census population projections for 2000 to 2050, the projected Hispanic population (in millions) can be modeled by the exponential function h(t) = 37.79(1.021)t, where t = 0 corresponds to 2000 and 0 ?t ? 50. Source: U.S. Census Bureau. a. Find the projected Hispanic population for 2005. Compare this to the actual value of 42.69 million. ________________ b. The U.S. Asian population is also growing exponentially, and the projected Asian population (in millions) can be modeled by the exponential function a(t) = 11.14(1.023)t, where t = 0 corresponds to 2000 and 0 ?t ? 50. Find the projected Asian population for 2005, and compare this to the actual value of 12.69 million. ________________ c. Determine the expected annual percentage increase for His-panics and for Asians. Which minority population, Hispanic or Asian, is growing at a faster rate? ________________ d. The U.S. black population is growing at a linear rate, and the projected black population (in millions) can be modeled by the linear function b(t) = 0.5116t + 35.43, where t = 0 corresponds to 2000 and 0 ?t ? 50. Find the projected black population for 2005 and compare this projection to the actual value of 37.91 million. ________________ e. Graph the projected population function for Hispanics and estimate when the Hispanic population will be double its actual value for 2005. Then do the same for the Asian and black populations. Comment on the accuracy of these numbers.

Problem 85A Drug Concentration When a pharmaceutical drug is injected into the bloodstream, its concentration at time t can be approximated by C(t) = C0e?kt, where C0 is the concentration at t= 0. Suppose the drug is ineffective below a concentration C1 and harmful above a concentration C2. Then it can be shown that the drug should be given at intervals of time T, where Source: Applications of Calculus to Medicine. A certain drug is harmful at a concentration five times the concentration below which it is ineffective. At noon an injection of the drug results in a concentration of 2 mg per liter of blood. Three hours later the concentration is down to 1 mg per liter. How often should the drug be given?

Problem 88A Evolution of Languages The number of years N(r) since two independently evolving languages split off from a common ancestral language is approximated by N (r) = 5000 ln r, where r is the proportion of the words from the ancestral language that are common to both languages now. Find the following. a.N (0.9) b.N (0.5) c.N (03) d. How many years have elapsed since the split if 70% of the words of the ancestral language are common to both languages today? e. If two languages split off from a common ancestral language about 1000 years ago, find r.

Problem 90A For Exercise, recall that log x represents the common (base 10) logarithm of x. Intensity of Sound The loudness of sounds is measured in a unit called a decibel. To do this, a very faint sound, called the threshold sound, is assigned an intensity I0. If a particular sound has intensity I, then the decibel rating of this louder sound is Find the decibel ratings of the following sounds having intensities as given. Round answers to the nearest whole number. a. Whisper, 115I0 ________________ b. Busy street, 9,500,000I0 ________________ c. Heavy truck, 20 m away, 1,200,000,000I0 ________________ d. Rock music concert, 895,000,000,000I0 ________________ e. Jetliner at takeoff, 109,000,000,000,000I0 ________________ f. In a noise ordinance instituted in Stamford, Connecticut, the threshold sound I0 was defined as 0.0002 microbars. Source: The New York Times. Use this definition to express the sound levels in parts c and d in microbars.

Problem 89A Communications Channel According to the Shannon-Hartley theorem, the capacity of a communications channel in bits per second is given by where B is the frequency bandwidth of the channel in hertz and s/n is its signal-to-noise ratio. Source: Scientific American. It is physically impossible to exceed this limit. Solve the equation for the signal-to-noise ratio s/n.

Problem 86A The graph for Exercise is plotted on a logarithmic scale where differences between successive measurements are not always the same. Data that do not plot in a linear pattern on the usual Cartesian axes often form a linear pattern when plotted on a logarithmic scale. Notice that on the horizontal scale, the distance from 5 to 10 is not the same as the distance from 10 to 15, and so on. This is characteristic of a graph drawn on logarithmic scales. Metabolism Rate The accompanying graph shows the basal metabolism rate (in cm3 of oxygen per gram per hour) for marsupial carnivores, which include the Tasmanian devil. This rate is inversely proportional to body mass raised to the power 0.25. Source: The Quarterly Review of Biology. a. Estimate the metabolism rate for a marsupial carnivore with body mass of 10 g. Do the same for one with body mass of 1000 g. ________________ b. Verify that if the relationship between x and y is of the form y = axb, then there will be a linear relationship between ln x and ln y. (Hint: Apply ln to both sides of y = axb.) ________________ c. If a function of the form y = axb contains the points (x1, y1) and (x2, y2), then values for a and b can be found by dividing by , solving the resulting equation for b, and putting the result back into either equation to solve for a. Use this procedure and the results from part a to find an equation of the form y = axb that gives the basal metabolism rate as a function of body mass. ________________ d. Use the result of part c to predict the basal metabolism rate of a marsupial carnivore whose body mass is 100 g.

Problem 91A For Exercise, recall that log x represents the common (base 10) logarithm of x. Intensity of Sound A story on the National Public Radio program All Things Considered, discussed a proposal to lower the noise limit in Austin, Texas, from 85 decibels to 75 decibels. A manager for a restaurant was quoted as saying, “If you cut from 85 to 75, … you’re basically cutting the sound down in half.” Is this correct? If not, to what fraction of its original level is the sound being cut? Source: National Public Radio.

Problem 93A For Exercise, recall that log x represents the common (base 10) logarithm of x. Acidity of a Solution A common measure for the acidity of a solution is its pH. It is defined by pH = ?log[H+], where H+ measures the concentration of hydrogen ions in the solution. The pH of pure water is 7. Solutions that are more acidic than pure water have a lower pH, while solutions that are less acidic (referred to as basic solutions) have a higher pH. a. Acid rain sometimes has a pH as low as 4. How much greater is the concentration of hydrogen ions in such rain than in pure water? ________________ b. A typical mixture of laundry soap and water for washing clothes has a pH of about 11, while black coffee has a pH of about 5. How much greater is the concentration of hydrogen ions in black coffee than in the laundry mixture?

Problem 92A For Exercise, recall that log x represents the common (base 10) logarithm of x. Earthquake Intensity The magnitude of an earthquake, measured on the Richter scale, is given by where I is the amplitude registered on a seismograph located 100 km from the epicenter of the earthquake, and I0 is the amplitude of a certain small size earthquake. Find the Richter scale ratings of earthquakes with the following amplitudes. a. 1,000,000I0 ________________ b. 100,000,000I0 ________________ c. On June 15, 1999, the city of Puebla in central Mexico was shaken by an earthquake that measured 6.7 on the Richter scale. Express this reading in terms of I0. Source: Exploring Colonial Mexico. ________________ d. On September 19, 1985, Mexico’s largest recent earthquake, measuring 8.1 on the Richter scale, killed about 10,000 people. Express the magnitude of an 8.1 reading in terms of I0. Source: History.com. ________________ e. Compare your answers to parts c and d. How much greater was the force of the 1985 earthquake than the 1999 earthquake? ________________ f. The relationship between the energy E of an earthquake and the magnitude on the Richter scale is given by where E0 is the energy of a certain small earthquake. Compare the energies of the 1999 and 1985 earthquakes. ________________ g. According to a newspaper article, “Scientists say such an earthquake of magnitude 7.5 could release 15 times as much energy as the magnitude 6.7 trembler that struck the North- ridge section of Los Angeles” in 1994. Source: The New York Times. Using the formula from part f, verify this quote by computing the magnitude of an earthquake with 15 times the energy of a magnitude 6.7 earthquake.