Base e and Natural

Logarithms

HistoryThe number e is a famous irrational number, and is one of the most important numbers in mathematics. The first few digits are2.7182818284590452353602874713

527...

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 1

Use a calculator to estimate to four decimal places.

0.5e 8eEx 2

Ex 3 Ex 4ln 3 1

ln4

Ex 5

Exponential logarithmicWrite an equivalent equation in the other

form.

23xe

Ex 6

Writing Equivalent Expressions

xe

Ex 8Ex 7

ln 1.2528x ln 2.25x

Inverse Properties

ln xe x ln xe x

Ex 9

Evaluate2 1ln xe

Ex 11

Writing Equivalent Expressions

Evaluate

ln 21e

Ex 10Evaluat

e

ln 3xe

Evaluate 7ln eEx 12

Solving Equations

23 4 10xe Ex 13

Solve the following equations.

22 5 15xe Ex 14

Solving Equations

Ex 15Solve the following equations.

Ex 16

ln 3 0.5x ln 3 3x

Graphing properties( and lnx are inverse

functions reflected in y=x)

These can be transformed (stretched, translated and reflected)in the same way as other functions. But, be careful of the hidden asymptotes!

xe

Examination-style questionThe function f is defined by

a) Describe the sequence of geometrical transformations by which the graph of y = 3ex + 1 – 4 can be obtained from that of y = ex.

b) The graph of y = f(x) crosses the y-axis at point A and the x-axis at point B. Write down the coordinates of A and B, working to 2 decimal places.

c) Write an expression for f –1(x) and state its domain and range.d) Sketch the graphs of y = f(x) and y = f –1(x) on the same set of axes

and state their geometrical relationship.

f 1( ) = 3 4xx e x

Examination-style questiond) y = f –1(x) is a reflection of y = f (x) in the line y = x.

y = x

x

y

y = –4

y = f(x)

y = f –1(x)

x = –4

(–0.71, 0)

(0, –0.71)

(0, 4.15)

(4.15, 0)