math 71b 9.3 – logarithmic functions 1. one-to-one functions have inverses. let’s define the...

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Math 71B

9.3 – Logarithmic Functions

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

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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

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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

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Logarithmic Form:

Exponential Form:

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Ex 1.Write each equation in the other form.

Exponential Form Logarithmic Form

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Ex 1.Write each equation in the other form.

Exponential Form Logarithmic Form

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Ex 1.Write each equation in the other form.

Exponential Form Logarithmic Form

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Ex 1.Write each equation in the other form.

Exponential Form Logarithmic Form

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Ex 1.Write each equation in the other form.

Exponential Form Logarithmic Form

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Ex 2.Evaluate.

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Note: ____ and ____

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

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Note: ____ and ____

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

1

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Note: ____ and ____

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

1 0

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Note: ____ and ____

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

1 0

𝒙

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Note: ____ and ____

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

1 0

𝒙 𝒙

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Ex 3.Evaluate.

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Ex 3.Evaluate.

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Ex 3.Evaluate.

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

Graph of Logarithmic Function

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

Graph of Logarithmic Function

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

Graph of Logarithmic Function

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

Graph of Logarithmic Function

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

Graph of Logarithmic Function

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

Graph of Logarithmic Function

ex: ex:

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Graph of Logarithmic Function

What is the vertical asymptote? _________

What is the domain? _________

What is the range? _________

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Graph of Logarithmic Function

What is the vertical asymptote? _________

What is the domain? _________

What is the range? _________

𝒙=𝟎

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Graph of Logarithmic Function

What is the vertical asymptote? _________

What is the domain? _________

What is the range? _________

(𝟎 ,∞ )

𝒙=𝟎

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Graph of Logarithmic Function

What is the vertical asymptote? _________

What is the domain? _________

What is the range? _________

(𝟎 ,∞ )

𝒙=𝟎

(−∞ ,∞)

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Ex 4.Graph Graph

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Ex 4.Graph Graph

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Ex 4.Graph Graph

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Ex 4.Graph Graph

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Ex 4.Graph Graph

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Ex 4.Graph Graph

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Ex 4.Graph Graph

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Ex 5.What is the domain of ?

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

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

common

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

commonnatural