Base e and Natural

Logarithms10.510.5

HistoryThe number e is a famous irrational number, and is one of the most important numbers in mathematics. The first few digits are2.71828182845904523536028747135

27...

It is often called Euler's number after Leonhard Euler. e is the base of the natural logarithms (invented by John Napier).

CalculatingThe value of (1 + 1/n)n approaches e as n gets bigger and bigger:

n (1 + 1/n)n

1 2.00000

2 2.25000

5 2.48832

10 2.59374

100 2.70481

1,000 2.71692

10,000 2.71815

100,000 2.71827

Vocabularynatural 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

11

n

n

Vocabularynatural 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

Ex 1Ex 1

Use a calculator to estimate to four decimal places.

0.5e 8eEx 2Ex 2

Ex 3Ex 3 Ex 4Ex 4ln 3 1

ln4

Ex 5Ex 5

Exponential logarithmicWrite an equivalent equation in the other form.

23xe Ex 6Ex 6

Writing Equivalent Expressions

6xe

Ex 8Ex 8Ex 7Ex 7

ln 1.2528x ln 2.25x

Inverse Properties

ln xe x ln xe x

Ex 9Ex 9

Evaluate2 1ln xe

Ex 11Ex 11

Writing Equivalent Expressions

Evaluate

ln 21eEx Ex 1010Evaluat

e

ln 3xe

Evaluate 7ln eEx 12Ex 12

Solving Equations

23 4 10xe Ex 13Ex 13

Solve the following equations.

22 5 15xe Ex 14Ex 14

Solving Equations

Ex 15Ex 15Solve the following equations.

Ex 16Ex 16

ln 3 0.5x ln 3 3x