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10.5 Base e and Natural Logarithms 10.5 Notes # ___

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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 2.7182818284590452353602874713527... 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 12.00000 22.25000 52.48832 102.59374 1002.70481 1,0002.71692 10,0002.71815 100,0002.71827

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Vocabulary natural base: the number e, which is found using It is the base rate of growth shared by all continually growing processes natural base exponential function: an exponential function with base e

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

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

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Ex 5 Write an equivalent logarithmic equation. Ex 6 Writing Equivalent Expressions Write an equivalent logarithmic equation.

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Ex 7 Write an equivalent exponential equation. Ex 8 Writing Equivalent Expressions Write an equivalent exponential equation.

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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

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Solving Equations Ex 14 Solve

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Solving Equations Ex 15 Solve

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Solving Equations Ex 16 Solve

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Solving Inequalities Ex 17 Solve

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Solving Inequalities Ex 18 Solve

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