1. Logarithmic and exponential relationships

2. Read: y = “log base b of a”
The inverse of an exponential function is a logarithmic function.
y = log b a
Read: y = “log base b of a”

3. logb A = x is read as “log base b of a equals X.
logb A = x if and only if bx = A.
The logarithm of a positive number A is the exponent to which the base must be raised to produce the number A.
logarithm: the exponent
Definition

4. EXAMPLES OF LOGARITHMS
log3 9 = 2 because 32 = 9
log2 8 = 3 because 23 = 8
log7 7 = 1 because 71 = 7
log5 25 = 2 because 52 = 25
log13 1 = 0 because 130 = 1
log2 ½ = -1 because 2-1 = ½
EXAMPLES OF LOGARITHMS

5. We can convert logarithmic equations to exponential equations and vice versa, using the following:
bx = A
x = logb A
Examples: #1 #2 #3
2x = 8
y -1 = 4
103 = 1000

6. LOGARITHMIC AND EXPONENTIAL FORMS
logb A = x is equivalent to bx = A.
Logarithmic Form
Exponential Form
logb X = a
ba = X
log3 9 = 2
32 = 9
log7 7 = 1
71 = 7
log13 1 = 0
130 = 1
LOGARITHMIC AND EXPONENTIAL FORMS

7. Converting to exponential form
If it starts out log, Use B.O.B – Bottom, Opposite, Back.
Example 1: log3 243 = 5
35 = 243
Converting to exponential form

8. Converting to exponential form
Use B.O.B – Bottom, Opposite, Back.
Example 2: log2 16 = 4
24 = 16
Converting to exponential form

9. Converting to exponential form
Use B.O.B – Bottom, Opposite, Back.
Example 3: log = 2
112 = 121
Converting to exponential form

10. Converting to exponential form
Use B.O.B – Bottom, Opposite, Back.
Example 4: logc T = a
ca = T
Converting to exponential form

11. Converting to logarithmic form
Write the word ‘log’.
Use B.A.E. – Base, Answer, Exponent.
Example 5: = 125
log5 125 = 3
Converting to logarithmic form

12. Converting to logarithmic form
Use B.A.E. – Base, Answer, Exponent.
Example 6: = 16
log4 16 = 2
Converting to logarithmic form

13. Converting to logarithmic form
Use B.A.E. – Base, Answer, Exponent.
Example 7: = 1
log12 1 = 0
Converting to logarithmic form

14. Converting to logarithmic form
Use B.A.E. – Base, Answer, Exponent.
Example 8: = ¼
log2 ¼ = -2
Converting to logarithmic form

15. Converting to logarithmic form
Use B.A.E. – Base, Answer, Exponent.
Example 9: Ca = B
logC B = a
Converting to logarithmic form