Math Logs Test Review Key

Algebra 2 Honors Logs Test Review Let f(x) = x², g(x) = √(x-5), and h(x) = (6-x)/3. Perform the indicated operation and state the domain when necessary. 1. g(h(6)) h(6) = (6-6)/3 = 0 g(0) = √(0-5) = √-5 (not possible) 2. h(g(3)) g(3) = √(3-5) = √-2 (not possible) 3. h(f(3)) f(3) = 3² = 9 h(9) = (6-9)/3 = -1 4. f(g(x)) f(√(x-5)) = (√(x-5))² = x-5 D: [5, ∞) 5. f(h^(-1)(x)) h(x) = (6-x)/3 3x = 6-y -3x+6 = y h^(-1)(x) = -3x+6 f(-3x+6) = (-3x+6)² = 9x²-36x+36 D: (-∞, ∞) 6. g(h(x)) g((6-x)/3) = √((6-x)/3-5) = √((6-x-15)/3) = √((-9-x)/3) D: (-∞, -9] 7. g(f(x)) g(x²) = √(x²-5) D: (-∞, -√5] U [√5, ∞) 8. h(f^(-1)(x)) f(x) = x² f^(-1)(x) = √x h(√x) = (6-√x)/3 D: [0, ∞) 9. h(g(x)) h(√(x-5)) = (6-√(x-5))/3 D: [5, ∞) Without graphing, determine whether or not the following functions have inverse functions. 10. f(x) = x²+17 No 11. f(x) = 2x-18+π Yes 12. f(x) = 5x^4+17x-8 No Find the following inverse functions, if they exist. 13. f(x) = x⁴/8 + 7 DNE 14. f(x) = 3/5x + 8 x = 3/5y + 8 5/3(x-8) = y f^(-1)(x) = 5/3(x-8) 15. f(x) = 2(x+5)^(3/2) x = 2(y+5)^(3/2) (x/2)^(2/3) = (y+5)^(3/2)^(2/3) (x/2)^(2/3) - 5 = y f^(-1)(x) = (x/2)^(2/3) - 5 Find the inverse function for each of the following: 16. f(x) = log₃x + 7 x = log₃y + 7 x - 7 = log₃y 3^(x-7) = y f^(-1)(x) = 3^(x-7) 17. g(x) = e^(x-4) x = e^(y-4) ln(x) = y - 4 ln(x) + 4 = y g^(-1)(x) = ln(x) + 4 18. h(x) = log₆(x+5) x = log₆(y+5) 6^x = y + 5 6^x - 5 = y h^(-1)(x) = 6^x - 5 Evaluate each of the following logarithms: 19. log₉27 = x 9^x = 27 3^(2x) = 3^3 2x = 3 x = 3/2 20. log₃√(27) = x 3^x = 27^(1/4) 3^x = (3^3)^(1/4) 3^x = 3^(3/4) x = 3/4 21. log₂(4√32)^3x = y 8^(3y) = (4√32)^(3x) 2^(3*3y) = (2^2 * 2^(5/2))^(3x) 2^(9y) = 2^(27/2 * x) 9y = 27/2 * x y = 9/2 22. log₁₀(1/10000) = v 10^v = 1/10000 10^v = 10^(-4) v = -4 Solve each of the following equations, if a solution exists: 23. log₈x = 5/2 8^(5/2) = x (2^3)^(5/2) = x 2^(15/2) = x 2√2^15 = x 24. log₉x = -3/2 9^(-3/2) = x (3^2)^(-3/2) = x 3^(-3) = x 1/27 = x 25. logₓ27 = 3/2 x^(3/2) = 27 (x^(3/2))^(2/3) = (27)^(2/3) x = (3^3)^(2/3) x = 9 26. logₓ125 = -1/2 x^(-1/2) = 125 (1/x)^(1/2) = 125 (1/125)^2 = x 1/15625 = x Condense the following logarithmic expressions into a single logarithm: 27. 4 - 2log₆a log₆1296 - log₆a² log₆(1296/a²) 28. 2log₃m - 1/2log₃n - 3 - log₃2 log₃m² - log₃√n - log₃3³ - log₃2 log₃(m²/(54√n)) Simplify the following logarithmic expressions: 29. log₅250 + log₅3 + log₅2 log₅(250 * 3 * 2) log₅1500 30. 1/6(2log₈4 + 2log₈2) 1/6(log₈4² + log₈2²) 1/6(log₈64) log₈√64 log₈2 Solve the following logarithmic equations. Check for extraneous solutions. Round answers to the nearest thousandth. 31. 2ln(x) = 7 ln(x) = 7/2 e^(7/2) = x x ≈ 33.115 32. log₄(3x) = log₄3 + log₄x log₄(3x) = log₄(3x) 3x = 3x All Real Numbers 33. log₄x - log₄(x-1) = 1/2 log₄(x/(x-1)) = 1/2 4^(1/2) = x/(x-1) 2 = x/(x-1) 2x - 2 = x x = 2 34. log₆(2x-5) - log₆(7x+10) = -1 log₆((2x-5)/(7x+10)) = log₆(1/6) (2x-5)/(7x+10) = 1/6 12x - 30 = 7x + 10 5x = 40 x = 8 35. log₂(x-10) + log₂(x+2) = 1 log₂((x-10)(x+2)) = 1 2^1 = x^2 + x - 20 x^2 + x - 22 = 0 x = (-1 ± √89)/2 x ≈ 4.22 or x ≈ -5.22 (extraneous) 36. ln(x) + ln(x+2) = 1 ln(x(x+2)) = 1 e^1 = x^2 + 2x x^2 + 2x - e = 0 x = (-1 ± √(1+4e))/2 x ≈ -2.718 or x ≈ 1.718 Solve the following exponential equations. You must have an exact answer. 37. 5^(2x) = 125^(2x-1) 5^(2x) = (5^3)^(2x-1) 5^(2x) = 5^(6x-3) 2x = 6x - 3 -4x = -3 x = 3/4 38. 81^(3-x) = (1/9)^(5x-6) * √27^x (3^4)^(3-x) = (3^-2)^(5x-6) * (3^(3/2))^x 3^(12-4x) = 3^(-10x+12+3x/2) 12 - 4x = -10x + 12 + 3x/2 -4x = -10x + 3x/2 6x = 3x/2 12x = 3x 9x = 0 x = 0 Solve the following exponential equations. Round answers to the nearest thousandth. 39. 2^(4x) = 2^(3-x) 4x = 3 - x 5x = 3 x = 0.6 40. e^(2x) = 7/4 ln(7/4) = 2x x ≈ 0.280 41. 12^x = 5^(x+4) log₁₂(5^(x+4)) = x (x+4)log₁₂5 = x (x+4)(0.6477) = x 0.6477x + 2.5907 = x 0.3523x = -2.5907 x ≈ -7.353 42. 3^(4+3x) = 8 3^(7x) = 8 log₃8 = 7x x ≈ 0.252 43. 5^(2x) = 4e^x 2xlog₅5 = log₅4 + log₅e^x 2x = log₅4 + x x = log₅4 ≈ 0.861 44. 100(1.04)^(2x) = 300 (1.04)^(2x) = 3 log₁₀(1.04)^(2x) = log₁₀3 2xlog₁₀1.04 = log₁₀3 x ≈ 14.006 45. x²2^x - 2^x = 0 2^x(x² - 1) = 0 2^x = 0 (no solution) or x² - 1 = 0 x = ±1 46. e^(4x) + 4e^(2x) - 21 = 0 Let u = e^(2x) u² + 4u - 21 = 0 (u + 7)(u - 3) = 0 u = -7 (not possible) or u = 3 e^(2x) = 3 2x = ln(3) x ≈ 0.549 47. x²e^(-3x) - 3x³e^(-3x) = 0 x³e^(-3x)(4 - 3x) = 0 x = 0 or x = 4/3 48. A man invests $5,000 in an account that pays 8.5% interest per year, compounded quarterly in July of 2008. a. Find the amount after 3 years. A = 5000(1 + 0.085/4)^(4*3) ≈ $6435.09 b. During which month of what year will the amount double? 2 = (1.02125)^(4t) log₁₀2 = 4tlog₁₀1.02125 t ≈ 8.241 years 8 years and 2.89 months September 2016 49. A man invests $6,500 in an account that pays 6% interest per year, compounded continuously. a. What is the amount after 2 years?   A = 6500e^(0.06*2) ≈ $7328.73 b. How long will it take for the amount to be $8,000? 8000 = 6500e^(0.06t) 1.230769 = e^(0.06t) ln(1.230769) = 0.06t t ≈ 3.46 years 50. During which month of what year will it take for an investment of $1,000 deposited in February of 1998 to double in value if the interest rate is 8.5% per year, compounded continuously?   2 = e^(0.085t) ln(2) = 0.085t t ≈ 8.1546 years 8 years and 1.85 months March 2006