Section 3.4 1. Find each logarithm. a. log 5 25 b . log 3 81 c . log 3 ⅓

1. Section 3.4 • 1. Find each logarithm. • a. log5 25 b. log3 81 c. log3 ⅓ • d. log3 1/9 e. log4 2 f. log2 ½ • log5 25 = 2 since 5 y = 25 = 5 2 implies y = 2. • log3 81 = 4 since 3 y = 81 = 3 4 implies y = 4. • log3 3-1 = -1 since 3 y = 3 - 1 implies y = - 1. • log3 3-2 = -2 since 3 y = 3 - 2 implies y = - 2. • log4 2 = ½ since 4 y = 2 = 4 1/2 implies y = 1/2. • log 2 1/2 = -1 since 2 y = 1/2 = 2 - 1 implies y = - 1.

2. Find each logarithm. • Use your calculator. • 2. a. ln (e10) b. ln √ec. ln3√e • d. ln 1 e. ln (ln(e e))f. lb (1/e 3 ) • = 10 • = ln e1/2 = ½ • = lne1/3= 1/3 • = 0 • = ln e because ln (ee) = e = 1 • = lne-3 = -3

3. 3. Use the properties of natural logarithms to simplify the following function. f(x) = ln (9x) – ln 9 Use the rule ln a – ln b = ln a/b. f(x) = ln (9x) – ln 9 = ln (9x/9) = ln x

4. 4. Use the properties of natural logarithms to simplify the following function. f(x) = ln (x 3) – ln x Use the rule ln x n = n ln x and the rule ln a – ln b = ln a/b. f(x) = ln (x 3 ) – ln x = 3 ln x – ln x = 2 ln x Or f (x) = ln (x 3/x) = ln (x 2 ) = 2 ln x.

5. 5. Use the properties of natural logarithms to simplify the following function f(x) = ln (x/4) + ln 4 Use the rule ln a + ln b = ln ab. f(x) = ln (x/4) + ln 4 = ln x – ln 4 + ln 4 = ln x OR f(x) = ln (x/4) + ln 4 = ln (x/4)(4) = ln x

6. 6. Use the properties of natural logarithms to simplify the following function f(x) = 8x – e lnx Use the rule e ln x = x. F(x) = 8x – e ln x = 8x – x = 7x

7. 7. Personal Finance: Interest - An investment grown at 24% compounded monthly. How many years will it take to • double? b. increase by 50%? • Use the formula • Since double M dollars is 2M dollars, we solve • M(1 + 0.24/12) 12n = 2M • M(1 + 0.02) 12n = 2M • 1.02 12n =2 then take the ln of both sides • ln (1.02) 12n = ln 2 • 12n ln 1.02 = ln 2 • CONTINUED

8. Answer to part b. • To find how many years it will take for the investment to increase by 50%: • M(1 + 0.24/12) 12n = 1.5M (Since an increase of 50% in M dollars is 1.5M.) • M(1 + 0.02) 12n = 1.5M • (1.02) 12n = 1.5 then take the ln of both sides • 12n ln (1.02) = ln 1.5 • Now, • A sum at 24% compounded monthly increases by 50% in about 1.7 years.

9. 8. Personal Finance: Interest - A bank offers 7% compound continuously. How soon will a deposit: • triple? b. increase by 25% • We use Pert with r = 0.07. Since triple P dollars is 3P, we solve • Pe0.07n = 3P • e 0.07n = 3 then take the ln of both sides • 0.07n = ln 3 • b. If P increases by 25%, the total amount is 1.25P. We solve • Pe0.07n = 1.25 P • e 0.07n = 1.25 then take the ln of both sides • 0.07n = ln 1.25

10. 9. Personal Finance: Depreciation - An automobile depreciates by 30% per year. How soon will it be worth only half its original value? [Hint: deprecation is like interest but at a negative rate.] If the deprecation is 30% = 0.30 per year, then r = -0.30. We use the interest formula P(1+r) n and we solve P(1 – 0.3) n = 0.5 P 0.7 n = 0.5 then take the ln of both sides ln 0.7 n = ln 5 n ln 0.7 = ln 5 and

11. 10. Business: Advertising - After t days of advertisements for a new laundry detergent, the proportion of shoppers in a town who have seen the ads is 1 – e -0.03t. How long must the ads run to reach 90% of the shoppers? We want to find the value of t that produces p(t) = 0.9. 0.9= 1 – e -0.03t e -0.03t = 0.1 then take the ln of both sides -0.03t = ln 0.1

12. 11. Behavioral Science: Forgetting - The proportion of students in a psychology experiment who could remember an eight-digit number correctly for t minutes was 0.9 – 0.2 ln t (for t >1) Find the proportion that remembered the number for 5 minutes. Simply replace t with 5 and calculate. p(5) = 0.9 – 0.2 ln 5 ≈ 0.58 or 58%

13. 12. General: Carbon 14 Dating - The proportion of carbon 14 still present in sample after t years is e -0.00012t. Estimate the age of the cave painting discovered which they were drawn contains only 2.3% of its original carbon 14. They are the oldest known paintings in the world. To find the number of years after which 2.3% = 0.023 of the origional carbon 14 is left, we solve, e -0.00012t.= 0.023 then take the ln of both sides -0.00012t = ln 0.023

14. 13. Social Science: education and income - According to a study, each additional year of education after high school graduation increases one’s income by 16%. Therefore, with x extra years of education, your income will be multiplied by a factor of 1.16 x. How many additional years of education are required to double your income? That is, find the x that satisfies 1.16 x = 2. 1.16 x= 2 then take the ln of both sides ln 1.16 x = ln 2 x ln 1.16 = ln 2

15. 14. Solve the following exercise on a graphing calculator by graphing an appropriate • exponential function together with a constant function and using INTERSECT to find the solution. • General: Inflation - At 2% inflation, prices increase by 2% compounded continuously. How soon will prices: • double? Use f (x) = e 0.02 t and y = 2 b. triple? Use f (x) = e 0.02 t and y = 3 • USE CALCULATOR • 35 years b. 55.5 years

16. 15. General: Carbon 14 Dating - In 1991 two hikers in the Italian Alps found the frozen but well preserved body of the most ancient human ever found, dubbed “Iceman”. Estimate the age of Iceman if his grass cape contained 53% of its original carbon 14. ( use the carbon 14 decay function, f (t) = e – 0.00012 t ) f (t) = e – 0.00012 t 0.53 = e – 0.00012 t then take the ln of both sides ln 0.53 = - 0.00012 t