Name________________________________________________________________________________Date_____________________Period______________A#_C-21_
Logarithms Investigation
PART I: Discovering Logarithms
Given below are the graphs of two exponential functions f and g.
1. Carefully sketch the inverse functions of each on the plots above. You can do this by switching the x and y coordinates (for
example, the point (0,1) becomes (1,0)). It may help you to first sketch the graph of 𝑦 = 𝑥.
a.
What is the domain and range of f(x)? What is the domain and range of its inverse, f-1(x)?
b.
What is the domain and range of g(x)? What is the domain and range of its inverse, g-1(x)?
2. For an exponential function ℎ(𝑥) = 𝑏 𝑥 , b is called the base of the exponential function. a.
From the graphs of f and g given above, how would one determine the approximate value of the base of each?
b.
Write plausible defining expressions for both f and g using this approximate value of the base.
𝑓(𝑥) =
𝑔(𝑥) =
3. Use your sketch to determine approximate values of 𝑓 −1 (2) and 𝑔−1 (2) where 𝑓 −1 and 𝑔−1 denote the inverse functions of f
and g respectively.
a.
𝑓 −1 (2) =
b.
Explain how you determined these values. What is a second way you could have found your answer? Verify
that this second method also works.
𝑔−1 (2) =
4. Explain why determining the value of 𝑓 −1 (2) is equivalent to asking the question: “what is x when 𝑓(𝑥) = 2?”
5. Think about the following question: “What is x when 10𝑥 = 36?” There are several ways to investigate this question. We
could begin by finding an interval that contains the solution. That is, we can easily calculate 101 = 10 and 102 = 100
and this would help us determine that the solution is between 1 and 2. By using our calculator to evaluate 101.5 = 31.6,
we could say that 1.5<x<2. We can continue this process to determine an approximate value for x. Use this method for
estimating the approximate value of x to the nearest hundredth for the following:
Estimates Using Guess-and-Check
6.
10𝑥 = 3
𝑥≈
10𝑥 = 10
𝑥=
10𝑥 = 36
𝑥≈
10𝑥 = 50
𝑥≈
10𝑥 = 100
𝑥=
10𝑥 = 125
𝑥≈
10𝑥 = 220
𝑥≈
Determine a graphical method for obtaining estimates for the solutions. Round to the nearest hundredth. Explain
your method and how you found a good window to view your graph. When you finish have a teacher initial it.
Teacher
Approval:
Estimates Using Graphing Technique
10𝑥 = 36
𝑥
𝑥≈
10 = 50
𝑥≈
10𝑥 = 3
𝑥≈
10𝑥 = 125
𝑥≈
10𝑥 = 220
𝑥≈
Explanation of method:
7. Using the LOG key, find each of the following values to the nearest hundredth.
log36= __________
log50= __________
log3=
__________
log125= __________
log 220 =__________
________________
8. Compare the values in the tables from Exercises 5, 6, and 7.
a.
Explain what the calculator displays when the LOG key is used.
b.
Without using your calculator, answer the following.
i. If 101.653 = 45, what is 𝑙𝑜𝑔45?
ii. If 𝑙𝑜𝑔132 = 2.121, what is 102.121 ?
iii. 𝑙𝑜𝑔104 =
iv. 10𝑙𝑜𝑔1000 =
9.
LOG on your calculator is short for common logarithm. Give the exponential function for which the common
logarithm function is the inverse function. Explain your reasoning and then request teacher approval.
Teacher
Approval:
________________
10.
Carefully draw the graph of 𝑓(𝑥) = 10𝑥 and 𝑓 −1 (𝑥) = log 𝑥 below. Use the table on your calculator to help you.
11.
Can a common logarithm of a real number be negative? If so, give an example. If not, explain why not.
12.
Do negative numbers have common logarithms? If so, give an example. If not explain why not.
Part II: Properties of Logarithms
1.
2.
Using calculator approximation methods from Part I question 5, fill in the table below, rounding to the nearest
hundredth. The notation 𝑔(𝑥) = 𝑙𝑜𝑔𝑏 𝑥 is used to denote the inverse function of the exponential function 𝑓(𝑥) = 𝑏 𝑥 .
If 3𝑥 = 36, then 𝑥 ≈
𝑙𝑜𝑔10 36 =
If 3𝑥 = 50, then 𝑥 ≈
𝑙𝑜𝑔10 50 =
If 3𝑥 = 3, then 𝑥 =
𝑙𝑜𝑔10 3 =
If 3𝑥 = 125, then 𝑥 ≈
𝑙𝑜𝑔10 125 =
If 3𝑥 = 220, then 𝑥 ≈
𝑙𝑜𝑔10 220 =
Teacher
Approval:
________________
Use your work from Part I and the table above to complete the following table. Round to the nearest hundredth:
𝑥
𝑙𝑜𝑔3 𝑥
𝑙𝑜𝑔10 𝑥
Ratio:
𝑙𝑜𝑔10 𝑥
𝑙𝑜𝑔3 𝑥
36
50
3
125
220
a.
Explain what you discovered about the ratio from the table’s third column.
b.
Examine the table again and identify the relationship among 𝑙𝑜𝑔10 𝑥, 𝑙𝑜𝑔3 𝑥, and 𝑙𝑜𝑔10 3. Write and equation to
express this relationship. Get teacher approval before proceeding.
Teacher
Approval:
________________
c.
The relationship found in part 2b does not only hold for the number 3 but for all numbers. Write you relationship
for any base b.
d.
Find 𝑙𝑜𝑔5 62 using your findings from part 2c.
3.
Fill in the following tables. Look back at other tables you filled in to save time. Round to the nearest Hundredth
3𝑥
𝑥
𝑙𝑜𝑔10 3
𝑙𝑜𝑔10 (3𝑥 )
𝑥 ∙ 𝑙𝑜𝑔10 3
36
50
3
125
220
a)
What do you immediately notice about the last two columns in the table?
3𝑥
36
6∙6
50
25 ∙ 2
3
3∙1
125
5 ∙ 25
220
20 ∙ 11
b)
4.
3𝑥 = 𝑐 ∙ 𝑑
𝑥
𝑙𝑜𝑔10 𝑐
𝑙𝑜𝑔10 𝑑
𝑙𝑜𝑔10 (𝑐 ∙ 𝑑)
𝑙𝑜𝑔10 𝑐 + 𝑙𝑜𝑔10 𝑑
What do you immediately notice about the last two columns in the table?
Summarize the relationships you noticed in parts 2c, 3a, and 3b by writing corresponding equations. (Hint – look at
the headers of the tables you used to answer the questions).
Equation for 2c:
Equation for 3a:
Equation for 3b:
5.
Using your results from question 4, make a guess at the Log rules for a, c, and d:
a. 𝑙𝑜𝑔𝑏 (𝑚 ∙ 𝑛) =
𝑚
b.
𝑙𝑜𝑔𝑏 ( ) = 𝑙𝑜𝑔𝑏 (𝑚) − 𝑙𝑜𝑔𝑏 (𝑛)
c.
𝑙𝑜𝑔𝑏 (𝑚𝑛 ) =
d.
(Change-of-Base formula) 𝑙𝑜𝑔𝑏 (𝑥) =
𝑛
Adapted from Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education
Coordinating Board. Available at http://www.utdanacenter.org/highered/alg2/downloads/IV-B-CourseContentAlgII/AlgII_6-1-2.pdf
6.
Copy your answers from question 5 to the space below to assist you on this page.
a.
7.
b.
c.
Using your log rules, complete the following equations using the respective rule.
a.
log 5 (21 ∗ 3) =
b.
log 7 (49 ) =
c.
log 5 ( ) =
d.
log 2 (𝑥) =
e.
log 9 (4) + log 9 (2) =
f.
log 9 (4) − log 9 (2) =
g.
log10 𝑥
h.
3 log 3 𝜋 =
99
3
log10 2
=
d.