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 )=bx, 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−1and g−1denote the inverse functions of f and g respectively.

a. f−1 (2 )=¿ g−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.

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 10 1.5 = 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

10x=3 x≈

10x=10 x=¿

10x=36 x≈

10x=50 x≈

10x=100 x=¿

10x=125 x≈

10x=220 x≈

6. 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.

Estimates Using Graphing Technique

10x=36 x≈

10x=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 =__________

Teacher Approval:

________________

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 log 45?

ii. If log 132=2.121, what is 102.121?

iii. log 104=¿

iv. 10log 1000=

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.

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.

Teacher Approval:

________________

Teacher Approval:

________________

Part II: Properties of Logarithms

1. 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 )=bx.

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≈ log10125=¿

If 3x=220, then x≈ log10220=¿

2. 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 xlog3 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 amonglog10 x, log3 x, and log10 3. Write and equation to express this relationship. Get teacher approval before proceeding.

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 log562 using your findings from part 2c.

Teacher Approval:

________________

3. Fill in the following tables. Look back at other tables you filled in to save time. Round to the nearest Hundredth

3x x log103 log10(3x) x ∙ log103

36

50

3

125

220

a) What do you immediately notice about the last two columns in the table?

3x 3x=c ∙d x log10 c log10d log10(c ∙d) log10 c+log10 d

36 6 ∙6

50 25 ∙2

3 3 ∙1

125 5 ∙25

220 20 ∙11

b) What do you immediately notice about the last two columns in the table?

4. 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. log b (m ∙n )=¿

b. log b(mn )=logb (m )−logb(n)

c. log b(mn)=¿

d. (Change-of-Base formula) log b ( x )=¿

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. b. c. d.

7. Using your log rules, complete the following equations using the respective rule.

a. log5(21∗3)=¿

b. log7(49)=¿

c. log5(993

)=¿

d. log2(x )=¿

e. log 9(4)+ log9(2)=¿

f. log 9(4)−log9(2)=¿

g.log10 xlog102

=¿

h. 3 log3π=¿