Math Chapter 7 Exponential and Logarithmic Functions

CHAPTER 7 - EXPONENTIAL AND LOGARITHMIC FUNCTIONS Determine whether each function represents exponential growth or decay. 1. y = 5(0.6)^x Decay 2. y = 0.1(2)^x Growth 3. y = 5.4^(-x) Decay Describe the shifts of the following exponential equations from its parent function and then graph. Then, identify the functions domain and range. 4. y = -2(1/2)^(x-3) Reflection over x-axis Right 3 D: All RN R: y < 0 5. y = 0.25(4)^(x-1) - 2 Left 1, Down 2 D: All RN R: y > -2 Write each expression in logarithmic form. 6. 2^7 = 128 log₂128 = 7 7. 8^(-2) = 1/64 log₈(1/64) = -2 Write each equation in exponential form. 8. log₈225 = 2 8² = 225 9. log₄32 = 5/2 4^(5/2) = 32 Write an equivalent exponential or logarithmic equation. 10. e^15 = x ln x = 15 11. ln 20 = x e^x = 20 12. e^(-2x) = 0.2 ln 0.2 = -2x 13. ln 0.0002 = x e^x = 0.0002 Evaluate each expression. 14. log₅25 2 15. log₄(1/64) -3 16. ln e⁷ 7 17. e³ e³ 18. ln e 1 19. log₃x = 5 3⁵ = x x = 243 20. log₅1/25 = x 5^x = 1/25 5^x = 5⁻² x = -2 21. log₆216 = x 6^x = 216 6^x = 6³ x = 3 22-25, use log₅3 ≈ 0.6826 and log₅4 ≈ 0.8614 to evaluate each expression. 22. log₅12 = log₅3 + log₅4 = 0.6826 + 0.8614 = 1.544 23. log₅(81/5) = log₅3⁴ - log₅5 = 4log₅3 - 1 = 4(0.6826) - 1 = 1.7304 24. log₅(9/16) = log₅3² - 2log₅4 = 2log₅3 - 2log₅4 = 2(0.6826) - 2(0.8614) = -0.3576 25. log₅144 = log₅3² + 2log₅4 = 2log₅3 + 2log₅4 = 2(0.6826) + 2(0.8614) = 3.088 Expand the following logarithms. 26. log₅4x² log₅4 + 2log₅x 27. ln(3x³/2y) ln3 + 3lnx - ln2 - lny Condense the following logarithms. 28. log₅24 - log₅6 log₅(24/6) log₅4 29. log₈6 + 2log₈3 log₈6 + log₈3² log₈6 + log₈9 log₈54 Solve each equation or inequality. Check your solution. 30. 3^(x-5) = 3⁴ x - 5 = 4 x = 9 31. 2^(3n-1) = (1/8)^(n) 2^(3n-1) = 2^(-3n) 3n - 1 = -3n 6n = 1 n = 1/6 32. 9^(2x-1) = 27^(x+4) 3^(2(2x-1)) = 3^(3(x+4)) 4x - 2 = 3x + 12 x = 14 33. e^x = 5 ln(e^x) = ln5 x = ln5 x = 1.609 34. 2e^x - 1 = 11 2e^x = 12 e^x = 6 ln(e^x) = ln6 x = ln6 x = 1.792 35. log₄(x+12) = 2 x + 12 = 4² x + 12 = 16 x = 4 36. logₓ(x + 3) = 2 x + 3 = x² 2 = x - 3 x = 5 37. log₅x + log₅x = log₅60 log₅x² = log₅60 5x² = 60 x² = 12 x = ±√12 38. log₄y + log₄8 = 1 log₄8y = 1 4¹ = 8y 4 = 8y y = 1/2 39. 3log₂3 - log₂4 = log₂b log₂3³ - log₂4 = log₂b log₂27/4 = log₂b 27/4 = b 40. ln 8 = 3x e³ = 8^x 20.0855 = 8^x x ≈ 2.511 41. ln(x+3) = 1 e¹ = x + 3 e - 3 = x 2.718 - 3 = x x ≈ -0.2817 42. y = 4^x and x = 4^y log₄x = y y = log₄x 43. y = ln(x+3) and x = ln(y+3) e^y = x + 3 y = e^x - 3 44. y = 8^x and x = log₈y y = 8^x x = log₈y 45. A computer system depreciates at a rate of 6.5% per year. If the computer system originally cost $4000, how long would it take for it to be worth half its value? 2000 = 4000(1 - 0.065)^t 0.5 = 0.935^t log 0.5 = t log 0.935 t ≈ 10.3 years 46. Suppose you deposit $1000 in an account paying 5% annual interest compounded continuously. a. What is the balance after 10 years? A = 1000e^(0.05*10) A = 1000(1.6487) A = $1648.72 b. How long will it take for the balance in your account to reach $1500? 1500 = 1000e^(0.05*t) 1.5 = e^(0.05*t) ln 1.5 = ln e^(0.05*t) ln 1.5 = 0.05t t ≈ 8.1 years 47. Suppose you deposit $100 in an account paying 3.5% interest compounded continuously. How long will it take for your money to double?   2 = e^(0.035t) ln 2 = ln e^(0.035t) ln 2 = 0.035t t = ln 2 / 0.035 ≈ 19.8 years 48. A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for half of this caffeine to be eliminated from a person's body?   1/2 = (1 - 0.11)^t log 1/2 = log 0.89^t log 1/2 = t log 0.89 t ≈ 5.9 hours 49. You're off to college! You buy a computer for $2500. It is expected to depreciate at a rate of 20% per year. What will be the value of the computer in 2 years?   y = 2500(1 - 0.20)^2 = 2500(0.8)^2 = $1600 50. A computer system depreciates at an average rate of 4% per month. If the value of the computer system was originally $12000, in how many months is it worth $7350? 7350 = 12000(1 - 0.04)^t 0.6125 = 0.96^t log 0.6125 = log 0.96^t t = log 0.6125 / log 0.96 ≈ 12 months 51. A piece of machinery valued at $250,000 depreciates at a fixed rate of 12% per year. After how many years will the value have depreciated to $100,000?   100000 = 250000(1 - 0.12)^t 0.4 = 0.88^t log 0.4 = log 0.88^t t = log 0.4 / log 0.88 ≈ 7.2 years 52. The Miller's bought a condominium for $185,000. Assuming that the value of the condo will appreciate at approximately 5% a year, how much will the condo be worth in 7 years?   y = 185000(1.05)^7 ≈ $260,313.58 53. The population of a city of one million people is increasing at a rate of 3% per year. If the population continues to grow at this rate, in how many years will the population have doubled?   2 = (1.03)^t log 2 = log 1.03^t log 2 = t log 1.03 t = log 2 / log 1.03 ≈ 23 years