8.7 – natural logarithms

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1.7 - Functions

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8.7 – Natural Logarithms. Natural Logarithm. Natural Logarithm – log e. Natural Logarithm – log e = ln. Natural Logarithm – log e = ln e. Natural Logarithm – log e = ln e ≈ 2.7183. Natural Logarithm – log e = ln e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e 2. - PowerPoint PPT Presentation

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Page 1: 8.7 – Natural Logarithms

1.7 - Functions

Page 2: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

Page 3: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

Page 4: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

Page 5: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

Page 6: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

There cannot be an x-value repeated!

Page 7: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

There cannot be an x-value repeated!

Page 8: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

There cannot be an x-value repeated!

Ex.1 Determine if each is a function.

Page 9: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

There cannot be an x-value repeated!

Ex.1 Determine if each is a function.

a. X Y

-6

-4 9

-1 -6

1 1

Page 10: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

There cannot be an x-value repeated!

Ex.1 Determine if each is a function.

a. X Y

-6

-4 9 Y

-1 -6 E

1 1 S

Page 11: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

There cannot be an x-value repeated!

Ex.1 Determine if each is a function.

a. X Y b.

-6

-4 9 Y

-1 -6 E

1 1 S

x y

-3 6

2 5

3 1

2 4

Page 12: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

There cannot be an x-value repeated!

Ex.1 Determine if each is a function.

a. X Y b.

-6

-4 9 Y

-1 -6 E

1 1 S

x y

-3 6

2 5

3 1

2 4

Page 13: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

There cannot be an x-value repeated!

Ex.1 Determine if each is a function.

a. X Y b.

-6

-4 9 Y

-1 -6 E

1 1 S

x y

-3 6

2 5

3 1

2 4

Page 14: 8.7 – Natural Logarithms

1.7 - Functions• A function is a relation in which each

element of the domain is paired with exactly one element of the range.

There cannot be an x-value repeated!

Ex.1 Determine if each is a function.

a. X Y b.

-6 NOT A

-4 9 Y FUNC.

-1 -6 E

1 1 S

x y

-3 6

2 5

3 1

2 4

Page 15: 8.7 – Natural Logarithms

Ex. 2 If f(x) = x2 – 5, find the following:

Page 16: 8.7 – Natural Logarithms

Ex. 2 If f(x) = x2 – 5, find the following:

a. f(-9)

Page 17: 8.7 – Natural Logarithms

Ex. 2 If f(x) = x2 – 5, find the following:

a. f(-9)

f(x) = x2 – 5

Page 18: 8.7 – Natural Logarithms

Ex. 2 If f(x) = x2 – 5, find the following:

a. f(-9)

f(x) = x2 – 5

f(-9)

Page 19: 8.7 – Natural Logarithms

Ex. 2 If f(x) = x2 – 5, find the following:

a. f(-9)

f(x) = x2 – 5

f(-9)

Page 20: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2

Ex. 2 If f(x) = x2 – 5, find the following:

Page 21: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

Ex. 2 If f(x) = x2 – 5, find the following:

Page 22: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

Ex. 2 If f(x) = x2 – 5, find the following:

Page 23: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

Ex. 2 If f(x) = x2 – 5, find the following:

Page 24: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

Ex. 2 If f(x) = x2 – 5, find the following:

Page 25: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

Ex. 2 If f(x) = x2 – 5, find the following:

Page 26: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) =

Ex. 2 If f(x) = x2 – 5, find the following:

Page 27: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

Ex. 2 If f(x) = x2 – 5, find the following:

Page 28: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62

Ex. 2 If f(x) = x2 – 5, find the following:

Page 29: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2

Ex. 2 If f(x) = x2 – 5, find the following:

Page 30: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2 – 5

Ex. 2 If f(x) = x2 – 5, find the following:

Page 31: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2 – 5

f(6z) = 36z2 – 5

Ex. 2 If f(x) = x2 – 5, find the following:

Page 32: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2 – 5

f(6z) = 36z2 – 5

c. f(4) + 2

Ex. 2 If f(x) = x2 – 5, find the following:

Page 33: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2 – 5

f(6z) = 36z2 – 5

c. f(4) + 2 f(4) + 2 =

Ex. 2 If f(x) = x2 – 5, find the following:

Page 34: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2 – 5

f(6z) = 36z2 – 5

c. f(4) + 2 f(4) + 2 = [(4)2 – 5]

Ex. 2 If f(x) = x2 – 5, find the following:

Page 35: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2 – 5

f(6z) = 36z2 – 5

c. f(4) + 2 f(4) + 2 =[(4)2 – 5]

Ex. 2 If f(x) = x2 – 5, find the following:

Page 36: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2 – 5

f(6z) = 36z2 – 5

c. f(4) + 2 f(4) + 2 =[(4)2 – 5] + 2

Ex. 2 If f(x) = x2 – 5, find the following:

Page 37: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2 – 5

f(6z) = 36z2 – 5

c. f(4) + 2 f(4) + 2 = [(4)2 – 5] + 2

= [16 – 5] + 2

Ex. 2 If f(x) = x2 – 5, find the following:

Page 38: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2 – 5

f(6z) = 36z2 – 5

c. f(4) + 2 f(4) + 2 = [(4)2 – 5] + 2

= [16 – 5] + 2 = 11 + 2

Ex. 2 If f(x) = x2 – 5, find the following:

Page 39: 8.7 – Natural Logarithms

a. f(-9)

f(x) = x2 – 5 f(-9) = (-9)2 – 5

= 81 – 5

f(-9) = 76

b. f(6z)

f(x) = x2 – 5

f(6z) = (6z)2 – 5

= 62·z2 – 5

f(6z) = 36z2 – 5

c. f(4) + 2 f(4) + 2 = [(4)2 – 5] + 2

= [16 – 5] + 2 = 11 + 2

f(4) + 2 = 13

Ex. 2 If f(x) = x2 – 5, find the following: