1. Exponential and Logarithmic Functions
Solving
Logarithm Properties
Inverses
Application
Graphing 10 20 30 40 50

2. Solve, round to nearest hundredth
5 2𝑥+8 = 125 𝑥
Answer

3. 5 2𝑥+8 = 125 𝑥
5 2𝑥+8 = 5 3𝑥
2𝑥+8=3𝑥
8=𝑥

4. Solve, round to nearest hundredth
7( 5 𝑥 )=168
Answer

5. 7( 5 𝑥 )=168
5 𝑥 =24
𝑥= log 5 24
𝑥= log 24 log 5
≈1.97

6. Solve, round to nearest hundredth
6 3𝑥 −20=3
Answer

7. 6 3𝑥 =23
3𝑥= log 6 23
3𝑥= log 23 log 6
3𝑥≈1.75
𝑥≈0.58

8. Solve, round to nearest hundredth
3+ log 4 (𝑥−7) =5
Answer

9. 3+ log 4 (𝑥−7) =5
log 4 (𝑥−7) =2
𝑥−7= 4 2
𝑥−7=16
𝑥=23

10. Solve, round to nearest hundredth
log (𝑥+3) − log 4 =3
Answer

11. log (𝑥+3) − log 4 =3
log 𝑥+3 4 =3
𝑥+3=4000
𝑥=3997
𝑥+3 4 = 10 3
𝑥+3 4 =1000

12. Write in logarithm form
𝑦= 7 𝑥
Answer

13. log 7 𝑦 =𝑥

14. Write in exponential form
𝑦= log 3 𝑥
Answer

15. 3 𝑦 =𝑥

16. Evaluate each of the expressions
log 18
log 5 17
log 4 64
Answer

17. log 18
≈1.256
log 5 17
≈1.760
log 4 64 =3

18. Simplify to a single logarithm
2 log 𝑎 −3 log 𝑏 +4 log 𝑐
Answer

19. 2 log 𝑎 −3 log 𝑏 +4 log 𝑐
log 𝑎 2 − log 𝑏 3 + log 𝑐 4
log 𝑎 2 𝑏 log 𝑐 4
log 𝑎 2 𝑐 4 𝑏 3

20. Expand the expression
log 2 𝑎 3 𝑏 4
Answer

21. log 2 𝑎 3 𝑏 4
log 2 𝑎 3 − log 𝑏 4
log 2 + log 𝑎 3 − log 𝑏 4
log 2 +3 log 𝑎 −4 log 𝑏

22. Find the inverse.
𝑦=( 5) 𝑥+3 −4
Answer

23. 𝑦=( 5) 𝑥+3 −4
𝑥=( 5) 𝑦+3 −4
𝑥+4=( 5) 𝑦+3
log 5 (𝑥+4) =𝑦+3
log 5 (𝑥+4) −3=𝑦

24. Find the inverse.
𝑦=7 (2) 𝑥+5
Answer

25. 𝑦=7 (2) 𝑥+5
𝑥=7 (2) 𝑦+5
log 2 𝑥 7 −5=𝑦
𝑥 7 = (2) 𝑦+5
log 2 𝑥 7 =𝑦+5

26. Find the inverse.
𝑦= log 8 𝑥−7
Answer

27. 𝑦= log 8 𝑥−7
𝑥= log 8 𝑦−7
𝑥+7= log 8 𝑦
8 𝑥+7 =𝑦

28. Find the inverse.
𝑦=4 log (3𝑥+7)
Answer

29. 𝑦=4 log (3𝑥+7)
𝑥=4 log (3𝑦+7)
10 𝑥 4 −7 3 =𝑦
𝑥 4 = log (3𝑦+7)
10 𝑥 4 =3𝑦+7
10 𝑥 4 −7=3𝑦

30. Find the inverse.
𝑦= 1 3 ln (𝑥+5) −2
Answer

31. 𝑦= 1 3 ln (𝑥+5) −2
𝑒 3(𝑥+2) =𝑦+5
𝑥= 1 3 ln (𝑦+5) −2
𝑒 3(𝑥+2) −5=𝑦
𝑥+2= 1 3 ln (𝑦+5)
3(𝑥+2)= ln (𝑦+5)

32. Suppose you deposit $1500 in a savings account that pays 6%
Suppose you deposit $1500 in a savings account that pays 6%. No money is added or withdrawn form the account.
Write an equation to model this situation.
How much will the account be worth in 5 years?
How many years until the account doubles?
Answer

33. Suppose you deposit $1500 in a savings account that pays 6%
Suppose you deposit $1500 in a savings account that pays 6%. No money is added or withdrawn form the account.
Write an equation to model this situation.
How much will the account be worth in 5 years?
How many years until the account doubles?
𝑦=1500 (1+.06) 𝑥
𝑦=1500 (1+.06) 5 =
3000=1500 (1+.06) 𝑥
12 years
𝑥= log =11.896

34. In 2009, there were 1570 bears in a wildlife refuge
In 2009, there were 1570 bears in a wildlife refuge. In 2010 approximately 1884 bears. If this trend continues and the bear population is increasing exponentially, how many bears will there be in 2018?
Write an exponential function to model the situation, then solve.
Answer

35. In 2009, there were 1570 bears in a wildlife refuge
In 2009, there were 1570 bears in a wildlife refuge. In 2010 approximately 1884 bears. If this trend continues and the bear population is increasing exponentially, how many bears will there be in 2018?
Write an exponential function to model the situation, then solve.
𝑦=𝑎 (𝑏) 𝑥
𝑦=1570 (1.2) 𝑥
𝑏= =1.2
𝑦=1570 (1.2) 9
8,100 bears

36. Suppose the population of a country is currently 7. 3 million people
Suppose the population of a country is currently 7.3 million people. Studies show this country’s population is declining at a rate of 2.3% each year.
Write an equation to model this situation.
How many years until the population goes below 4 million?
Answer

37. Suppose the population of a country is currently 7. 3 million people
Suppose the population of a country is currently 7.3 million people. Studies show this country’s population is declining at a rate of 2.3% each year.
Write an equation to model this situation.
How many years until the population goes below 4 million?
𝑃=7.3 (1−0.023) 𝑡
4=7.3 (1−0.023) 𝑡
𝑡= log (0.5479) =25.854
26 years

38. 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 t, where a is the amount of carbon-14 originally in the object, and t is the age of the object in years.
A fossil of a bone contains 32% of its original carbon-14. What is the approximate age of the bone?
Answer

39. 𝑦=𝑎 𝑒 − 𝑡
32=100 𝑒 − 𝑡
0.32= 𝑒 − 𝑡
ln 0.32 =− 𝑡
ln − =𝑡
𝑡=9,496 years

40. A new truck that sells for $29,000 depreciates 12% each year
A new truck that sells for $29,000 depreciates 12% each year. What is the value of the truck after 7 years?
Answer

41. 𝑦=29000 (1−0.12) 𝑥
𝑦=29000 (1−0.12) 7
𝑦=11,851.59
$11,851.59

42. Graph and Identify the domain and range
𝑦= 2 𝑥−2 −3
Answer

43. 𝑦= 2 𝑥−2 −3
Domain: All real numbers
Range: 𝑦>−3

44. Graph and Identify the domain and range
𝑦=2 2 𝑥−3 +1
Answer

45. 𝑦=2 2 𝑥−3 +1
Domain: All real numbers
Range: 𝑦>1

46. Graph and Identify the domain and range
𝑦= log 3 (𝑥+1) +2
Answer

47. 𝑦= log 3 (𝑥+1) +2
Domain: 𝑥>−1
Range: All real numbers

48. Graph and Identify the domain and range
𝑦=2 log 5 (𝑥) −3
Answer

49. 𝑦=2 log 5 (𝑥) −3
Domain: 𝑥>0
Range: All real numbers

50. Graph and Identify the domain and range
𝑦=−3 2 𝑥+1 +2
Answer

51. 𝑦=−3 2 𝑥+1 +2
Domain: All real numbers
Range: 𝑦<2