Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Write each expression as a single natural logarithm.

11. 3 ln 5
12. ln 9 + ln 2
13. ln 24 − ln 6 ln 24 minus ln 6
14. 5 ln m − 3 ln n 5 ln m minus 3 ln n
15. 1 3 ( ln x + ln y ) − 4 ln z 1 third , open ln x plus ln y close minus 4 ln z
16. l n a − 2 ln b + 1 3 ln c l n eh minus 2 ln b plus , 1 third , ln c
17. 4 ln 8 + ln 10
18. ln 3 − 5 ln 3 ln 3 minus 5 ln 3
19. 2 ln 8 − 3 ln 4 2 ln 8 minus 3 ln 4

See Problem 2.

Solve each equation. Check your answers.

20. l n 3 x = 6 l n 3 x equals 6
21. l n x = − 2 l n x equals negative 2
22. l n ( 4 x − 1 ) = 36 l n open 4 x minus 1 close equals 36
23. 1.1 + ln x 2 = 6 1.1 plus ln , x squared , equals 6
24. l n x − 1 2 = 4 l n . fraction x minus 1 , over 2 end fraction . equals 4
25. l n 4 r 2 = 3 l n , 4 r squared , equals 3
26. 2 ln 2 x 2 = 1 2 ln , 2 x squared , equals 1
27. l n ( 2 m + 3 ) = 8 l n open 2 m plus 3 close equals 8
28. ln ( t − 1 ) 2 = 3 ln . open t minus 1 close squared . equals 3

See Problem 3.

Use natural logarithms to solve each equation.

29. e x = 18 e to the x , equals 18
30. e x 5 + 4 = 7 e super x over 5 end super , plus 4 equals 7
31. e 2 x = 12 e super 2 x end super , equals 12
32. e x 2 = 5 e super x over 2 end super , equals 5
33. e x + 1 = 30 e super x plus 1 end super . equals 30
34. e 2 x = 10 e super 2 x end super , equals 10
35. e 3 x + 5 = 6 e super 3 x end super , plus 5 equals 6
36. e x 9 − 8 = 6 e super x over 9 end super , minus 8 equals 6
37. 7 − 2 e x 2 = 1 7 minus 2 , e super x over 2 end super , equals 1

See Problem 4.

Space For Exercises 38 and 39, use v = − 0 . 0098 t + c ln R , v equals negative 0 . , 0098 to the t , plus c ln r comma  where v is the velocity of the rocket, t is the firing time, c is the velocity of the exhaust, and 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.

    C 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 = a e − 0.00012 t , y equals . eh e super negative , 0.00012 , t end super . comma 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?
41. Archaeology A fossil bone contains 25% of its original carbon-14. What is the approximate age of the bone?

Simplify each expression.

42. ln 1
43. l n e 4 fraction l n e , over 4 end fraction
44. l n e 2 2 fraction l n , e squared , over 2 end fraction
45. l n e 83 l n , e to the eighty third
46. l n e l n e
47. l n e 2 l n , e squared
48. l n e 10 l n , e to the tenth
49. 10 ln e
50. l n e 3 l n , e cubed
51. l n e 4 8 fraction l n , e to the fourth , over 8 end fraction

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