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**Base e and Natural Logarithms 10.5**

Notes # ___ 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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**The value of (1 + 1/n)n approaches e as n gets bigger and bigger: n**

Calculating The value of (1 + 1/n)n approaches e as n gets bigger and bigger: n (1 + 1/n)n 1 2 5 10 100 1,000 10,000 100,000

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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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**Use a calculator to estimate to four decimal places.**

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

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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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**Use a calculator to estimate to four decimal places.**

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

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**Writing Equivalent Expressions**

Write an equivalent logarithmic equation. Ex 6 Write an equivalent logarithmic equation.

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**Writing Equivalent Expressions**

Write an equivalent exponential equation. Ex 8 Write an equivalent exponential equation.

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Inverse Properties

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**Writing Equivalent Expressions**

Evaluate Evaluate Ex 11 Ex 12 Evaluate Evaluate

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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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LOGS EQUAL THE The inverse of an exponential function is a logarithmic function. Logarithmic Function x = log a y read: “x equals log base a of y”

LOGS EQUAL THE The inverse of an exponential function is a logarithmic function. Logarithmic Function x = log a y read: “x equals log base a of y”

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