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**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**

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

Ex 1 Ex 2 Ex 3 Ex 4

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**Writing Equivalent Expressions**

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

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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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**Solve the following equations.**

Solving Equations Solve the following equations. Ex 13 Ex 14

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**Solve the following equations.**

Solving Equations Solve the following equations. Ex 15 Ex 16

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Essential Question: What are some of the similarities and differences between natural and common logarithms.

Essential Question: What are some of the similarities and differences between natural and common logarithms.

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