# Base e and Natural Logarithms 10.5

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Base e and Natural Logarithms 10.5

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).

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

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

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

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

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

Inverse Properties

Writing Equivalent Expressions
Evaluate Evaluate Ex 11 Ex 12 Evaluate Evaluate

Solve the following equations.
Solving Equations Solve the following equations. Ex 13 Ex 14

Solve the following equations.
Solving Equations Solve the following equations. Ex 15 Ex 16