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Math 71B 9.3 – Logarithmic Functions 1

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Page 1: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Math 71B

9.3 – Logarithmic Functions

Page 2: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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One-to-one functions have inverses. Let’s define the inverse of the exponential function.

Page 3: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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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

Page 4: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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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

Page 5: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Logarithmic Form:

Exponential Form:

Page 6: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 1.Write each equation in the other form.

Exponential Form Logarithmic Form

Page 7: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 1.Write each equation in the other form.

Exponential Form Logarithmic Form

Page 8: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 1.Write each equation in the other form.

Exponential Form Logarithmic Form

Page 9: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 1.Write each equation in the other form.

Exponential Form Logarithmic Form

Page 10: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 1.Write each equation in the other form.

Exponential Form Logarithmic Form

Page 11: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 2.Evaluate.

Page 12: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Note: ____ and ____

Note: Since and are inverse functions by definition, _____ and _____

Page 13: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Note: ____ and ____

Note: Since and are inverse functions by definition, _____ and _____

1

Page 14: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Note: ____ and ____

Note: Since and are inverse functions by definition, _____ and _____

1 0

Page 15: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Note: ____ and ____

Note: Since and are inverse functions by definition, _____ and _____

1 0

𝒙

Page 16: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Note: ____ and ____

Note: Since and are inverse functions by definition, _____ and _____

1 0

𝒙 𝒙

Page 17: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 3.Evaluate.

Page 18: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 3.Evaluate.

Page 19: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 3.Evaluate.

Page 20: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Let’s graph , knowing that it’s the inverse of :

Graph of Logarithmic Function

Page 21: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Let’s graph , knowing that it’s the inverse of :

Graph of Logarithmic Function

Page 22: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Let’s graph , knowing that it’s the inverse of :

Graph of Logarithmic Function

Page 23: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Let’s graph , knowing that it’s the inverse of :

Graph of Logarithmic Function

Page 24: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Let’s graph , knowing that it’s the inverse of :

Graph of Logarithmic Function

Page 25: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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In general, the logarithmic function with base is (where , , ).

Graph of Logarithmic Function

ex: ex:

Page 26: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Graph of Logarithmic Function

What is the vertical asymptote? _________

What is the domain? _________

What is the range? _________

Page 27: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Graph of Logarithmic Function

What is the vertical asymptote? _________

What is the domain? _________

What is the range? _________

𝒙=𝟎

Page 28: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

28

Graph of Logarithmic Function

What is the vertical asymptote? _________

What is the domain? _________

What is the range? _________

(𝟎 ,∞ )

𝒙=𝟎

Page 29: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Graph of Logarithmic Function

What is the vertical asymptote? _________

What is the domain? _________

What is the range? _________

(𝟎 ,∞ )

𝒙=𝟎

(−∞ ,∞)

Page 30: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 4.Graph Graph

Page 31: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 4.Graph Graph

Page 32: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 4.Graph Graph

Page 33: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

33

Ex 4.Graph Graph

Page 34: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

34

Ex 4.Graph Graph

Page 35: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 4.Graph Graph

Page 36: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 4.Graph Graph

Page 37: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Ex 5.What is the domain of ?

Page 38: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Notation: (_____________ logarithm) (_____________ logarithm) Ex 6.

Page 39: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Notation: (_____________ logarithm) (_____________ logarithm) Ex 6.

common

Page 40: Math 71B 9.3 – Logarithmic Functions 1. One-to-one functions have inverses. Let’s define the inverse of the exponential function. 2

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Notation: (_____________ logarithm) (_____________ logarithm) Ex 6.

commonnatural


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