DO NOT WRITE ON THIS SHEET!
You are EXPECTED TO TAKE NOTES IN YOUR NOTEBOOK! A copy of this lesson will be posted on Mrs. Ghillany’s web site if you want to print a copy for yourself.
Logarithms are simply another way to write exponents . Exponential and logarithmic functions are inverses of each other.
Logarithms are the "opposite" of exponentials , just as subtraction is the opposite of addition and division is the opposite of multiplication
Definition of Logarithmic Function
:
—The Relationship—
y = b x
..............is equivalent to...............
(means the exact same thing as) log b
(y) = x log b
(y) = x because
y = b x
(Read as “log base b of y equals x”) b is the base in the exponential equation, and it is the base of the logarithm as well.
Exponential function is solved for ‘y’ and logarithmic function is solved for ‘x’
Answer to exponential equation is the outcome/ final answer , but the answer to logarithmic equation is the exponent ‘x’ !
CONVERTING BETWEEN LOGS AND EXPONENTIAL EXPRESSIONS
:
Convert the following functions into either exponential or logarithmic form:
Example 1 : Change each logarithmic expression to an exponential expression.
1. log
3
27 = 3 2. log
36
6 = 1 / 2 3. log
2
(1 / 8) = -3 4. log
8
2 = 1 / 3
TURN OVER!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Solutions
1. The logarithmic form log
3
27 = 3 is equivalent to the exponential form 27 = 3 3
2. The logarithmic form log
36
6 = 1 / 2 is equivalent to the exponential form 6 = 36 1/2
3. log
2
(1 / 8) = -3 in exponential form is given by 1 / 8 = 2 -3
4. log
8
2 = 1 / 3 in exponential form is given by 2 = 8 1/3
********************************************************************************************
Example 2 : Change each exponential expression to logarithmic expression.
1. 3 4 = 81 2. 4 1/2 = 2 3. 3 -3 = 1 / 27 4. 10 3 = 1000
*************************************************************************************************************
Solutions
1. The exponential form 3 4 = 81 is equivalent to the logarithmic form 4 = log
3
81
Since 3 is the base and 4 is the exponent, we will have log base 3 of 81 equal to 4.
2. The exponential form 4 1/2 = 2 is equivalent to the logarithmic form 1 / 2 = log
4
2
3. 3 -3 = 1 / 27 in logarithmic form is given by -3 = log
3
(1/ 27)
4. 10 3 = 1000 in logarithmic form is given by 3 = log
10
1000
Evaluating Logs with the Definition of Logarithm
:
Logarithm with base 10 such as log
10
( x ) is called common logarithm and is written as log x
(base 10 is “invisible”) Common Logarithm can be evaluated on the calculator. NOTE: LOG key on your calculator indicates common log with base 10 only! If you press LOG on your calculator, calculator is using 10 as the base!
********************************************************************************************************************************************
Examples: Evaluate the following functions using the definition of logs.
1.
f x
 log 32 2. f x
 log 1
3
3. f x
 log 2
4
4. f x
 log
10
1
100
5. log
5
(25). 6. log
2
(8). 7. log
64
(4). 8. log
6
(6). 9. log
4
(–16). 10. log
2
(0).
SOLUTIONS. NOTE: ANSWER TO LOGARITHM is the POWER/EXPONENT !
1. f x
 log 32 base = 2
2 to what power is 32?
2. f x
 log 1
3
3.
base = 3
3 to what power is 1? f x
 log 2
4
4. f x
 log
10
1
100 base = 4 base = 10
4 to what power is 2? 10 to what power is 1/100? log
2
32

5 log
3
1

0 log
4
2

1
2
5. log
5
(25). 6. log
2
(8). 7. log
64
(4). 8. log
6
(6). 9. log
4
(–16).
1 log
10
100
 
2
10. log
2
(0). log
5
25

2 log
2
8

3 log
64
4

1
3 log
6
6

1 log
4
(

16 )
 undefined log
2
0
 undefined
It is not possible t raise base to power and get an answer < 0!!! Log argument must be >0!!!

DO NOT WRITE ON THIS SHEET!
You are EXPECTED TO TAKE NOTES IN YOUR NOTEBOOK! A copy of this lesson will be posted on Mrs. Ghillany’s web site if you want to print a copy for yourself.
Important Logarithmic Identities (rules that are always true and are VERY
IMPORTANT to understand/remember)
:
WITH YOUR PARTNER, CAREFULLY GO OVER THE FOLLOWING IDENTITIES AND COME UP
WITH EXAMPLES USING NUMBERS, TO UNDERSTAND WHY THESE RULES ARE ALWAYS TRUE! log (
 a
1 ) 0 because a
0 
1 log a
( a )

1 because a
1  a log a
( a x
)
 x because a x  a x log a
( 0 ) is undefined log a
( 0 )
 n there is no way we can raise a positive base to some power
and end up with zero. log a
( x )
 log a
( y ) a x  a y
 x = y
********************************************************************************************************************
EVALUATING LOGARITHMS THAT DO NOT SIMPLIFY EASILY
Last class, yesterday, you learned how to convert between exponential expressions and logarithmic expressions, and you also learned how to evaluate simple logarithms. For example:
1. f x
 log 32 base = 2
2 to what power is 32? log
2
32

5
2. f x
 log 1
3
3.
base = 3
3 to what power is 1? log
3
1

0 f x
 log 2
4
4. f x
 log
10
1
100 base = 4 base = 10
4 to what power is 2? 10 to what power is 1/100? log
4
2

1
2
HOWEVER, NOT ALL LOGARITHMS SIMPLIFY EASILY.
1 log
10
100
 
2
For instance, try to evaluate log 8
.
Base is 3. 3 to what power is equal to 8??? The answer/exponent here is not an integer value.
There is a formula that you can use with logarithms that do not simplify easily.
TURN OVER!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

a

b

b x

a x
a b
The argument of the logarithm stays the same ‘x’ for both, but we change both logs into the logarithms with common base. The best base value to use is base = 10, because any calculator can evaluate common logarithm with base 10. a =10
Examples: Evaluate the following expressions, using the change of base formula. a. log 8
3 b. log 10 c. log 15
9 log
3
8
 log 8 log 3

1 .
89 log
12
10
 log 10 log 12

0 .
93 log
9
15
 log 15 log 9

1 .
23
Hints: base value goes to the denominator “bottom” of the formula using base 10 makes it easy to evaluate on the calculator make sure you close the parentheses at the end of the argument of each logarithm always give one or two decimal places for your final answer! loglogloglog loglogloglogloglogloglogloglogloglogloglogloglogloglogloglogloglogloglogloglogloglog
Mixed Practice: Evaluate the following expressions/equations, using the definitions and properties of logs.
1.
7
2. log 5
5 x
5. log
7
 1 
 49 
6. 
3
4
9. log
7 x
 log 9
7
10. log 5
1
=x
2
13. log 4
3 14. log 64
8
17.
2 log
5
3 x

4
18.
2
3 x

81
16
3.
7. 3 x 
4
11. log
8 x
 log 10

1
15. 3 ( 5 ) x 
7
19. e x 
0
4. log
10
1
 
8. log
10
1
1000
12.

1
16.
2
3 log x

6
20.
2
3
 x 
565
Expect THESE problems on your
!!!
OTL provides additional practice and answers!