Math Analysis - Chapter 5.4, 5.5, 5.6 Mathlete: Date Assigned Section Homework (due the next day) Mon 4/17 Tue 4/18 5.4 Properties of Logarithms P.413: 7,9,13,15,17,19,21,23,25,27,31,33,37,41,43,45 5.5 Exponential and Logarithmic Equations p.421: 5-29 odd, 35-55 odd Wed 4/19 Thu 4/20 5.5 Exponential and Logarithmic Equations (Applications) 5.6 Exponential, Logistic, and Logarithmic Models p.422: 61, 63, 67, 69, 73, 75, 77 p.432: 7-25 odd, 31, 35, 37 Fri 4/21 Review 5.4, 5.5, 5.6 Review Sheet 5.4-5.6 Mon 4/24 Tue 4/25 Review 5.4, 5.5, 5.6 Quiz 5.4, 5.5, 5.6 p.442: 49-95 odd 1
5.4 Properties of Logarithms Product Property of Logarithms Example 1: Use the Product Property to calculate the logarithm without a calculator. Then check your answer with a calculator. Given log 2.5 0.3979 and log 3 0.4771, calculate: a. log 25 b. l og 75 Given log2 0.3010 and log5 0.6990, and log7 0.8451 calculate: 5 8. log 14 b. og l 7 c. log 5 Power Property of Logarithms Example 2: Simplify the following expression, if possible, by eliminating exponents and radicals. 3 a. log(xy ) b. ln(3x 1/2 3 y) c. (lnx) 1/3 Quotient Property of Logarithms 3 x 20. log y 2 2
Write the 3 properties of logarithms from the previous page: 1. 2. 3. Write each logarithm as a sum and/or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. x y 22. log 5 4 l 4 y 3 e 5 1000 24. n 26. og 3 x y+1 l a a 4 28. log a z5 xy 4 Write each expression as a logarithm of a single quantity and then simplify if possible. 32. l og 4.1 + log 3 34. ln y ln 2 + ln x 38. 3 log x + 2 42. ln(x 2 9) l n(x + 3 ) 1 44. 2[log(x 2 9) log(x 3)] logx 46. log 16x log y 3 2 4 1 8 2 3
5.5 Exponential and Logarithmic Equations One-to-One Property For a>0, and a 1, If a x = a y, then x = y In other words, if you can write the equation with the same bases, then you can equate the exponents. Example 1: Solve the Exponential Equation (using one-to-one property) 2 t = 128 7 2x = 49 6 x = 1 216 Example 2: Solve the Exponential Equation (taking the log or ln of both sides) 1. Isolate the base and exponent. 2. Then take the log or ln of both sides. 12. 5e x = 60 14. 3 x x = 7 16. 6(0.9 ) = 7 18. 3 x = 10 4x+1 24. 4e x + 6 = 22 28. 5 + e x 2 +1 = 8 4
You cannot take the log of a negative number. For example: log( 5 ) is undefined. Not possible! To find an extraneous solution, plug your answers back into the original equation. If you find that you are taking the log of a negative number, then the corresponding solution is extraneous. Example 3: Solve the logarithmic equation and eliminate any extraneous solutions. 36. l n x = 1 40. log(x 2 ) = 3 44. l og(x + 3 ) + log(x 3 ) = 0 48. log5 x = 1 log 5(x 4 ) 52. l og(x + 5) log(4x 2 + 5 ) = 0 54. l og(3x + 1 ) + l og(x + 1 ) = 1 5
5.5 Exponential and Logarithmic Equations (Applications) Continuous Compound Interest Determine how long it takes for the given investment to double if r is the interest rate and the interest is compounded continuously. 62. Initial amount $3000, r = 4% 64. Initial amount $6000, r = 6.25% Find the interest rate r if the interest on the initial deposit is compounded continuously. 68. Initial amount $3000, amount in 3 yrs: $3600 70. Initial amount $6000, amount in 10 yrs: $12,000 6
74. Bacterial Growth Suppose the population of a colony of bacteria doubles in 20 hours from an initial population of 1 million. Find the growth constant k if the population is modeled by the function P (t) = P e kt 0. When will the population reach 4 million? 8 million? 76. Depreciation The value of a 2003 Toyota Corolla is given by the function v (t) = 14, 000(0.93) where t is the number of years since its purchase and v (t) is its value in dollars. a. What was the Corolla s initial purchase price? b. What percent of its value does the Corolla lose each year? c. How long will it take for the value of the Corolla to reach $12,000? t 7
5.6 Exponential, Logistic, and Logarithmic Models Let s do a summary of exponential functions. Exponential Growth f (x) a > 0 b > 1 = a b x asymptote: Exponential Decay f (x) a > 0 = a b x 0 < b < 1 asymptote: 30. The population of the US is expected to grow from 282 million in 2000 to 335 million in 2020. a. Find a function of the form P (t) = Ce kt that models the population growth of the US. Here, t is the number of years since 2000 and P(t) is in millions. b. In what year will the population be 300 million? 8
The Logistic Model (p.430) In the previous section, we examined population growth models using an exponential function. However, it seems unrealistic that any population would tend to infinity over a long period of time. Other factors, such as the environment to support the population would eventually come into play and level off the population. Thus we need a more refined model of population growth that takes such issues into account. One function is called the logistic function : c f (x) = 1+ae bx where a, b, and c are constants. As x increases, the function values f(x) approach c because the denominator approaches 1. As x decreases, the function values f(x) approach 0 because the denominator increases in magnitude. 36. Health Sciences The spread of the flu in an elementary school can be modeled by a logistic function. The number of children infected with the flu virus t days after the first infection is given by N (t) = 150. 1+4e 0.5t a. How many children were initially infected with the flu? b. How many children were infected with the flu virus after 5 days? After 10 days? 9
Practice Problems: Use f (t) = 10e t. 8. Evaluate f(2). 10. For what value of t will f(t)=2? Use f (t) e. = 4 t 12. Evaluate f(3) 14. For what value of t will f(t)=10? 10 Use f (x) = 1+2e 0.5x 16. Evaluate f(1). 18. Evaluate f(12). Use f (x) = 3lnx 4 20. Evaluate f(1). 22. For what value of x will f(x)=3? 10