Name______

Unit 6:Launching Task: Historical relevence and overview of properties

Investigating the Properties of Logarithms

**For the purpose of this activity we will be using Common Logs.

PART I

1.  Complete the following table using your calculator. Round answers to four decimal places.

log 5 / 0.6990
log 10 – log 2
log 3
log 18 – log 6
log 7
log 28 –log 4
log ½
Log 3 – log 6
log 2
Log 8 – log 4

2.  Using any patterns you see in the results above, what generalizations could be made?

3.  How could we find the value of log 3 if the “3” button is missing from our calculator? Explain.

PART II

4.  Complete the following table using your calculator. Round answers to four decimal places.

log 12
log 6 + log 2
log 18
log 3 + log 6
log 9
log 3 + log 3
log 20
log 4 + log 5
log 26
log 2 + log 13

5.  Using any patterns you see in the results above, what generalizations could be made?

6.  How could we find the value of log 30 if the “3” button is missing from our calculator? Explain.

PART III

7.  Complete the following table.

Number / Equivalent value with a different base
16 / 24, 42
64
81
49
25

8.  Complete the following table using your calculator. Round values to four decimal places.

4log 2 / 1.2041
log 16
2log 5
log 25
3log 4
6log 2
log 64
2log 7
log 49

9.  Using the two previous tables, what generalization(s) can be made? Can these generalizations be linked to your previous knowledge of exponents? How?

10.  A student noticed that log ½ gave the same value as –log 2. How is this possible?

Summarize all the properties of logarithms you know. Compare your results with others in the class.

Definition of an exponent.

If x is any real number and n is any positive integer, by definition:

(a) if n = 1 then

(b) if then

(n factors of x)

we call the nth power of x. The integer n is called the exponent of to the base x.

Exponential Functions

The exponential function f with base a is denoted by

where and x is any real number.

Theorems or Rules of exponents

1. 

2. 

3. 

4. 

5. 

6.  By definition
Let then,
by exponent rules 1 and 3

if the two numbers are equal, and they have the same base then their exponents must also be equal.

therefore

7.  By definition
Let then,
by exponent rules 1 and 3

if the two numbers are equal, and they have the same base then their exponents must also be equal.

therefore

Notes on Definition of a Logarithm Name ______

The logarithmic function is the inverse function of the exponential function.

If
then

is called the logarithmic function of x to the base a.

Rules or Theorems of Logarithms

1. 

2. 

3. 

4.  If

5.  Let
then by definition of a logarithm
Multiplying
by the first rule of exponents
by definition of a logarithm
by transitive law of equality.

6.  Let
then by definition of a logarithm
Dividing
by the second rule of exponents
by definition of a logarithm
by transitive law of equality.

7.  If
and by rule 5 of exponents
by definition of a logarithm

Classwork/ Homework: Pg 512 (13-27 odd)