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10.5 Base e and Natural Logarithms 10.5

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History The number e is a famous irrational number, and is one of the most important numbers in mathematics. The first few digits are It is often called Euler's number after Leonhard Euler. e is the base of the natural logarithms (invented by John Napier).

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Calculating The value of (1 + 1/n) n approaches e as n gets bigger and bigger: n (1 + 1/n) n , , ,

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Vocabulary natural base: the number e, which is found using the base rate of growth shared by all continually growing processes Used heavily in science to model quantities that grow & decay continuously natural base exponential function: an exponential function with base e

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Vocabulary natural logarithm: a logarithm with base e The natural log gives you the time needed to reach a certain level of growth. natural logarithmic function: the inverse of the natural base exponential function

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Ex 1 Use a calculator to estimate to four decimal places. Ex 2 Ex 3 Ex 4

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Ex 5 Exponential logarithmic Write an equivalent equation in the other form. Ex 6 Writing Equivalent Expressions Ex 8 Ex 7

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Inverse Properties

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Ex 9 Evaluate Ex 11 Writing Equivalent Expressions Evaluate Ex 10 Evaluate Ex 12

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Solving Equations Ex 13 Solve the following equations. Ex 14

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Solving Equations Ex 15 Solve the following equations. Ex 16

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