Properties of Logarithms 2

Write as a single logarithm.

1. 3loga x +12loga y − 3loga z 2. log2 x − log2 6 + 2 log2 5 − log8 27

Solve for x.

3. log5 6 − log5 (x − 2) = log5 3 4. 6 log27 x = 1

5. 36−x = 42+x 6. x = 52 log5 6

7. 5x2 −x = 7 8. log2(x + 4) + log2(x + 2) = log2 3

9. erx = 2 10. log x4( ) = logx( )3

Miscellaneous.

11. If log2 3 = a , what is log8116 in terms of a?

12. Solve for x: ln(ln(ln(x+6))) = 0

13. Evaluate. Remember, do NOT use a calculator.

a) log24 85

2 b) log2 56 − log8

343

c) 34 log3 5 d) log7 16

log7116

14. If loga b = M , find loga a2b2( ) in terms of M.

15. Solve for x. 3x+1 = 5x−1 Only use your calculator once at the last step!

16. Find the inverse of the following function and identify its domain. f(x) = 2x+1 − 3

17. The atmospheric pressure varies with altitude above the surface of the Earth. Meteorologists have determined that for altitudes up to 10 km, the pressure, p, in millimeters of mercury is given by p = 760e−0.125a , where a is the altitude in km.

a) What is the atmospheric pressure at an altitude of 3.3 km?

b) At what altitude will the atmospheric pressure be 450 mm of mercury?

18. log 4x( )− log(12+ x) = 2 19.

4001 + e−x = 350

20. Graph y = log3(−x − 4) . Identify domain and range.

21. Find an equation of an exponential function y= f(x) that passes through (3, 9) and (4, 28) and evaluate f(-1).

22. The value of a house appreciates (that is, grows) at a rate of 2.5% each year. Today the house is worth $ 350,000.

a. Provide a model for the value of the house t years from today.b. When will the house be worth $ 550,000?

23. The half-life of Carbon 14 is 5730 years. c. What percentage of Carbon 14 would you expect in a piece of wood 12,000 years

old?d. How old is a mammoth tooth that contains 45% of its original carbon content?

24. How many years will it take you to double your investment of 4,000 if your bank is offering a 4% annual rate compounded twice each month?

25. An experiment begins with approximately 400 bacteria, which grow at such a rate that the population triples every hour. a) Find the bacteria population 1.5 hours after the experiment began.

b) After how many hours the bacteria population will reach 50,000?

26. Prove the following “Product of Three Logs Rule”:

(loga b) ⋅ (logb c) ⋅ (logc d) = loga d

27. Use the Reciprocal Rule for Logs, loga b =1

logb a (easily proved with the Change of Base

formula), to solve the following system of equations:

log9 x + logy 8 = 2

logx 9 + log8 y =83

28. What annual interest rate compounded monthly would assure that your investment of $ 4,000 will double in 10 years?

29. Solve for x.

a) 3d =m

tre−2x + B b) log7 (log5 (log2 | x |)) = 0

c) 5x +12(5− x ) = 7 d) log(10x + 5) − log(x − 4) = log(2)

30. A house purchased 4 years ago for $240,000 was sold today for $260,000. IF the value of the house continues to increase exponentially at the same rate, how much will the house sell for in 8 years?

Prove the following properties. Justify each step.

31. loga b =

1logb a

32. loga b( ) logb c( ) = loga c

33. log

bn x =1n

logb x 34. logbn xn = logb x

35. log1

b

1x= logb x

36. Solve for y: log5 x( ) logx 5x( ) log5x y( ) = log1

x

x−4

37. Given log 2 = a , log3 = b , and log7 = c , find log(1i2i3i4i5i6i7i8i9i10) in terms of a, b,

and c.

38. Find an equation of an exponential function that passes through points (3, 4) and (5, 18). Does it also pass through the point (-1, 1/9)? Prove your answer.

39. All of the logarithmic and exponential work we have been doing could be done with just one base. Mathematically speaking, what she meant is expressed by the following law:

logb x = c ⋅ loga x, where c is a constant.

If Geoff ’s favorite base is b = 15.8, what constant c should he be using in converting the logarithms presented with Andrew’s favorite base, a = 2 ? Give an expression for the exact value of c and give a reasonable estimate of the value of c.

40. Assume that the number of hours that milk stays fresh decreases exponentially with temperature. Suppose that milk in the refrigerator at 0°C will keep for 192 hours, and milk left out in the kitchen at 20°C will keep for only 48 hours.

a) Write the equation expressing the number of hours milk keeps as a function of the temperature.

b) What temperature (to nearest degree) is necessary for milk to be kept for 90 hours?

41. a) Give 4 different possible equations for the graph below, 2 exponential and 2 logarithmic.

b) Find the equation of the inverse of this function.

42. Solve for x. log 5 – log (2x) = 4

43. Solve for x. logx 81 = 3

44. Given: logb a = 0.898244 , logb c = −1, and ab = 30,

a) find loga b b) find logb ac2( )

c) find the values of a, b and c.

45. What exponential function of the form y = aebx goes through the points (3, 10) and (6, 50)?

46. Let f(x) = 3x . Find f−1(1) .

47. Solve

a)

�

log 3x + 9( ) − log x +1( ) = log x −1( ) b) log 2 log9 81( ) = x

48. Farah makes a tuna fish sandwich for lunch at 6 AM. She takes the usual sanitary precautions, but her dog, Fido, comes and licks her hand, and 8 bacteria leap onto the sandwich. If the amount of bacteria grows by 12 percent every 20 minutes, how many bacteria will grace the tuna sandwich by lunch at noon?

49. Solve for y:

�

log1y

(2x + 8) − log1y

(2x 2 +10x + 8) = −1

50. Solve for x. log16 log 2 log256 x( )⎡⎣ ⎤⎦ =

12

51. Prove that

log(x + h) − logxh

= log 1 + hx

⎛⎝⎜

⎞⎠⎟

1h

.

52. Prove that loga x( ) logb y( ) = logb x( ) loga y( ) .

53. If f(x) = log2 x , evaluate f 165

4⎛

⎝⎜

⎞

⎠⎟ .

54. State the domain and range of y = log5( x − 63 ) .

55. Solve (no calc)

log5 x + log25 x = log.2 3

56. a) What could the question have been to a bank interest problem if the correct solution involves solving the equation:

10,000e.055t = 10,000 1+ .0612

⎛⎝⎜

⎞⎠⎟12t

?

b) Solve the equation.

57. A house bought four years ago for $40,000 was just sold for $60,000. If the value of the house continues to increase exponentially at the same rate, how much will it be worth in 10 years from now?

58. On the same day, Nick deposits $500 into account A that pays 10% interest compounded yearly, and $1000 into account B that earns 6% interest compounded quarterly. a) Sketch a graph of both situations on the same set of axes. You may use your graphing calculator to help you.

b) After how many years will the accounts have the same balance?

59. Solve for x. x2(logx 27)(log9 x) = x + 4

60. Solve for x. loga x + loga2 x + log

a3 x = 11