math 71b 9.3 – logarithmic functions 1. one-to-one functions have inverses. let’s define the...
DESCRIPTION
3 Logarithmic FunctionTRANSCRIPT
1
Math 71B
9.3 – Logarithmic Functions
2
One-to-one functions have inverses. Let’s define the inverse of the exponential function.
3
To find inverse of , we first switch and : Then we solve for .
We don’t have tools to solve for , so we just define what’s called the _________________________:
(Note: Here, and are positive, and .)
Logarithmic Function
4
To find inverse of , we first switch and : Then we solve for .
We don’t have tools to solve for , so we just define what’s called the _________________________:
(Note: Here, and are positive, and .)
Logarithmic Function
logarithmic function
5
Logarithmic Form:
Exponential Form:
6
Ex 1.Write each equation in the other form.
Exponential Form Logarithmic Form
7
Ex 1.Write each equation in the other form.
Exponential Form Logarithmic Form
8
Ex 1.Write each equation in the other form.
Exponential Form Logarithmic Form
9
Ex 1.Write each equation in the other form.
Exponential Form Logarithmic Form
10
Ex 1.Write each equation in the other form.
Exponential Form Logarithmic Form
11
Ex 2.Evaluate.
12
Note: ____ and ____
Note: Since and are inverse functions by definition, _____ and _____
13
Note: ____ and ____
Note: Since and are inverse functions by definition, _____ and _____
1
14
Note: ____ and ____
Note: Since and are inverse functions by definition, _____ and _____
1 0
15
Note: ____ and ____
Note: Since and are inverse functions by definition, _____ and _____
1 0
𝒙
16
Note: ____ and ____
Note: Since and are inverse functions by definition, _____ and _____
1 0
𝒙 𝒙
17
Ex 3.Evaluate.
18
Ex 3.Evaluate.
19
Ex 3.Evaluate.
20
Let’s graph , knowing that it’s the inverse of :
Graph of Logarithmic Function
21
Let’s graph , knowing that it’s the inverse of :
Graph of Logarithmic Function
22
Let’s graph , knowing that it’s the inverse of :
Graph of Logarithmic Function
23
Let’s graph , knowing that it’s the inverse of :
Graph of Logarithmic Function
24
Let’s graph , knowing that it’s the inverse of :
Graph of Logarithmic Function
25
In general, the logarithmic function with base is (where , , ).
Graph of Logarithmic Function
ex: ex:
26
Graph of Logarithmic Function
What is the vertical asymptote? _________
What is the domain? _________
What is the range? _________
27
Graph of Logarithmic Function
What is the vertical asymptote? _________
What is the domain? _________
What is the range? _________
𝒙=𝟎
28
Graph of Logarithmic Function
What is the vertical asymptote? _________
What is the domain? _________
What is the range? _________
(𝟎 ,∞ )
𝒙=𝟎
29
Graph of Logarithmic Function
What is the vertical asymptote? _________
What is the domain? _________
What is the range? _________
(𝟎 ,∞ )
𝒙=𝟎
(−∞ ,∞)
30
Ex 4.Graph Graph
31
Ex 4.Graph Graph
32
Ex 4.Graph Graph
33
Ex 4.Graph Graph
34
Ex 4.Graph Graph
35
Ex 4.Graph Graph
36
Ex 4.Graph Graph
37
Ex 5.What is the domain of ?
38
Notation: (_____________ logarithm) (_____________ logarithm) Ex 6.
39
Notation: (_____________ logarithm) (_____________ logarithm) Ex 6.
common
40
Notation: (_____________ logarithm) (_____________ logarithm) Ex 6.
commonnatural