Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Write each expression as a single logarithm.

9. log 7 + log 2 log 7 plus log 2
10. log 2 9 − log 2 3 log base 2 , 9 minus , log base 2 , 3
11. 5 log 3 + log 4 5 log 3 plus log 4
12. log 8 − 2 log 6 + log 3 log 8 minus 2 log 6 plus log 3
13. 4 log m − log n 4 log m minus log n
14. log 5 − k log 2 log 5 minus k log 2
15. log 6 5 + log 6 x log base 6 , 5 plus , log base 6 , x
16. log 7 x + log 7 y − log 7 z log base 7 , x plus , log base 7 , y minus , log base 7 , z
17. log 3 4 + log 3 y + log 3 8 x log base 3 , 4 plus , log base 3 , y plus , log base 3 . 8 to the x

See Problem 2.

Expand each logarithm.

18. log x 3 y 5 log , x cubed , y to the fifth
19. log 7 49 x y z log base 7 , 49 x y z
20. log b b x log base b . b over x
21. log a 2 log , eh squared
22. log 5 r s log base 5 . r over s
23. log 3 ( 2 x ) 2 log base 3 , open , 2 to the x , close 2
24. log 3 7 ( 2 x − 3 ) 2 log base 3 , 7 open 2 x minus 3 close 2
25. log a 2 b 3 c 4 log . fraction eh squared , b cubed , over c to the fourth end fraction
26. log 4 5 x log base 4 , 5 square root of x
27. log 8 8 3 a 5 log base 8 , 8 , square root of 3 , eh to the fifth end root
28. log 5 25 x log base 5 . 25 over x
29. log 10 m 4 n − 2 log . 10 m to the fourth . n super negative 2 end super

See Problem 3.

Use the Change of Base Formula to evaluate each expression.

30. log 2 9 log base 2 , 9
31. log 12 20 log base 12 , 20
32. log 7 30 log base 7 , 30
33. log 5 10 log base 5 , 10
34. log 4 7 log base 4 , 7
35. log 3 54 log base 3 , 54
36. log 5 62 log base 5 , 62
37. log 3 33 log base 3 , 33
38. Science The concentration of hydrogen ions in household dish detergent is 10 − 12 . 10 super negative 12 end super . .  What is the pH level of household dish detergent?

B Apply

See Problem 4.

Use the properties of logarithms to evaluate each expression.

39. log 2 4 − log 2 16 log base 2 , 4 minus , log base 2 , 16
40. log 2 96 − log 2 3 log base 2 , 96 minus , log base 2 , 3
41. log 3 27 − 2 log 3 3 log base 3 , 27 minus 2 , log base 3 , 3
42. log 6 12 + log 6 3 log base 6 , 12 plus , log base 6 , 3
43. log 4 48 − 1 2 log 4 9 log base 4 , 48 minus , 1 half . log base 4 , 9
44. 1 2 log 5 15 − log 5 75 1 half . log base 5 , 15 minus , log base 5 , square root of 75
45. Think About a Plan The loudness in decibels (dB) of a sound is defined as 10 log I I 0 , 10 log . fraction i , over i sub 0 end fraction . comma  where I is the intensity of the sound in watts per square meter ( W/m 2 ) . I 0 , open , cap wslashm squared , close . , i sub 0 , comma  the intensity of a barely audible sound, is equal to 10 − 12 W/m 2 . 10 super negative 12 end super . cap wslashm squared , . Town regulations require the loudness of construction work not to exceed 100 dB. Suppose a construction team is blasting rock for a roadway. One explosion has an intensity of 1.65 × 10 − 2 1.65 , times , 10 super negative 2 end super   W/m 2 . cap wslashm squared , . Is this explosion in violation of town regulations?
    • Which physical value do you need to calculate to answer the question?
    • What values should you use for I and I 0 ? i sub 0 , question mark
46. Construction The foreman of a construction team puts up a sound barrier that reduces the intensity of the noise by 50%. By how many decibels is the noise reduced? Use the formula L = 10 log I I 0 l equals 10 log . fraction i , over i sub 0 end fraction  to measure loudness. (Hint: Find the difference between the expression for loudness for intensity I and the expression for loudness for intensity 0. 5 I . 5 to the i , . )
47. Error Analysis Explain why the expansion at the right of log 4 t s log base 4 . square root of t over s end root  is incorrect. Then do the expansion correctly.
48. Reasoning Can you expand log 3 ( 2 x + 1 ) ? log base 3 , open 2 x plus 1 close question mark  Explain.

49. Writing Explain why log ( 5 · 2 ) ≠ log 5 · log 2 . open 5 middle dot 2 close not equal to log 5 middle dot log 2 .

    

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