Pg 481 #12-36 even, 42-50 even please help! With work :)

Practice and Problem-Solving Exercises C MATHEMATICAL PRACTICES Practice Write each expression as a single natural logarithm. See Problem 1. 11. 3 In 5 12. In 9 + In 2 13. In 24 - In 6 14. 5 In m - 3 Inn 15. = (In x + Iny) - 4Inz 16. In a - 2 Inb + =Inc 17. 4 In 8 + In 10 18. In 3 - 5 In 3 19. 2 In 8 - 3 In 4 Solve each equation. Check your answers. See Problem 2. 20. In 3x = 6 21. In x = -2 22. In (4x - 1) = 36 23. 1.1 + In x2 = 6 24. In * = 1 = 4 25. In 472 = 3 26. 2 In 2x2 = 1 27. In (2m + 3) = 8 28. In (t - 1)2 = 3 Use natural logarithms to solve each equation. See Problem 3. 29. et = 18 30. e5 + 4 = 7 31. e2x = 12 32. e2 = 5 33. ex+1 = 30 34. e2x = 10 35. e3x + 5 = 6 36. es - 8 = 6 37. 7 - 2e2 = 1 STEM Space For Exercises 38 and 39, use v = -0.0098t + c In R, where u is the velocity of the rocket, t is the firing time, c is the velocity of the exhaust, and See Problem 4. R is the ratio of the mass of the rocket filled with fuel to the mass of the rocket without fuel. 38. Find the velocity of a spacecraft whose booster rocket has a mass ratio of 20, an exhaust velocity of 2.7 km/s, and a firing time of 30 s. Can the spacecraft achieve a stable orbit 300 km above Earth? 39. A rocket has a mass ratio of 24 and an exhaust velocity of 2.5 km/s. Determine the minimum firing time for a stable orbit 300 km above Earth. Apply 40. Think About a Plan By measuring the amount of carbon-14 in an object, a paleontologist can determine its approximate age. The amount of carbon-14 in an object is given by y = ae -0.000121, where a is the amount of carbon-14 originally in the object, and t is the age of the object in years. In 2003, a bone believed to be from a dire wolf was found at the La Brea Tar Pits. The bone contains 14% of its original carbon-14. How old is the bone? . What numbers should you substitute for y and t? . What properties of logarithms and exponents can you use to solve the equation? STEM 41. Archaeology A fossil bone contains 25% of its original carbon-14. What is the approximate age of the bone? Simplify each expression. 42. In 1 43, Ine 44, In e2 4 45. In e83 46. In e 47. In e2 48. In e10 49. 10 Ine 50. In e3 51. In et 8