8.7 – Natural Logarithms

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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

Transcript of 8.7 – Natural Logarithms

  • 1.7 - Functions

  • 1.7 - FunctionsA function is a relation in which each element of the domain is paired with exactly one element of the range.

  • 1.7 - FunctionsA function is a relation in which each element of the domain is paired with exactly one element of the range.

  • 1.7 - FunctionsA function is a relation in which each element of the domain is paired with exactly one element of the range.

  • 1.7 - FunctionsA function is a relation in which each element of the domain is paired with exactly one element of the range.

  • 1.7 - FunctionsA 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!

  • 1.7 - FunctionsA 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!

  • 1.7 - FunctionsA 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.

  • 1.7 - FunctionsA 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.X Y -6-4 9-1 -6 1 1

  • 1.7 - FunctionsA 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.X Y -6-4 9 Y-1 -6 E 1 1 S

  • 1.7 - FunctionsA 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.X Y b. -6-4 9 Y-1 -6 E 1 1 S

  • 1.7 - FunctionsA 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.X Y b. -6-4 9 Y-1 -6 E 1 1 S

  • 1.7 - FunctionsA 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.X Y b. -6-4 9 Y-1 -6 E 1 1 S

  • 1.7 - FunctionsA 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.X Y b. -6NOT A -4 9 YFUNC.-1 -6 E 1 1 S

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

  • Ex. 2 If f(x) = x2 5, find the following: a. f(-9)

  • Ex. 2 If f(x) = x2 5, find the following: a. f(-9) f(x) = x2 5

  • Ex. 2 If f(x) = x2 5, find the following: a. f(-9) f(x) = x2 5 f(-9)

  • Ex. 2 If f(x) = x2 5, find the following: a. f(-9) f(x) = x2 5 f(-9)

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5

  • Ex. 2 If f(x) = x2 5, find the following: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: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: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:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) =

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2 5

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2 5 f(6z) = 36z2 5

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2 5 f(6z) = 36z2 5 c. f(4) + 2

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2 5 f(6z) = 36z2 5 c. f(4) + 2 f(4) + 2 =

  • Ex. 2 If f(x) = x2 5, find the following:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2 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:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2 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:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2 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:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2 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:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2 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:a. f(-9) f(x) = x2 5 f(-9) = (-9)2 5 = 81 5 f(-9) = 76 b. f(6z) f(x) = x2 5f(6z) = (6z)2 5 = 62z2 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