The Natural Log Function

5.1 Differentiation

5.2 Integration

A Brief History of e• 1616-1618 John Napier, Scottish, Inventor of

Logarithms, e implied in his work—gave table of natural log values (although did not recognize e as base)

• 1661 Christian Huygens, Dutch, studies relationship between the area under a rectangular hyperbola and logarithms, but does not see connection to e (later he does evaluate log e to 17 decimal places, but does see that this is a log)

• 1668 Nicolaus Mercator, German, names the natural logarithm, but does not discuss its base

A Brief History of e

• 1683 Jacques Bernoulli, Swiss, discovers e through study of compound interest, does not call it e or recognize its connections to logs

• 1697 Johann Bernoulli (Jacques brother) begins study of the calculus of exponential functions and is perhaps the first to recognize logs as functions

• 1720s Leonard Euler, Swiss, first studied e, proved it irrational, and named it (the fact that it is the first letter of his surname is coincidental).

A Brief History of Logs

• Napier: studies the motion of someone covering a distance d whose speed at each instant is equal to the remaining distance to be covered. He divided the time into short intervals of length  , and assumed that the speed was constant within each short interval. He tabulated the corresponding values of distance and time obtained in this way.

• He coined a name for their relationship out of the Greek words logos (ratio) and arithmos (number). He used a Latinized version of his word: logarithm.

• In his table, Napier chose = 10-7 (and d = 107). In modern terms, we can say that the base of the logarithm in Napier's table was

• The actual concept of a base was not developed until later.

Definition of Natural Log Function

• 1647--Gregorius Saint-Vincent,

Flemish Jesuit, first noticed that

the area under the curve from 1 to e is 1, but does not define or recognize the importance of e.

Definition of Natural Log Function

0 ,1

ln1

xdtt

xx

x

y

x

y

Definition of e

67182818284.2

11

ln1

e

dtt

ee

Review: Properties of Logs(i.e. Making life easier!)

ana

bab

a

baab

n lnln

lnlnln

lnln)ln(

0)1ln(

Practice: Expand each expression

1

3ln

5

6ln

23ln

9

10ln

23

22

xx

x

x

x

Derivatives of the Natural Log Function

u

u

dx

du

uu

dx

d

uu

u

dx

du

uu

dx

d

xx

xdx

d

'1ln

0 ,'1

ln

0 ,1

ln

Practice: Find f ’ (Don’t forget your chain rule!!)

)1ln()( )2

)2ln()( )1

2

xxf

xxf

3ln)( )4

ln)( )3

xxf

xxxf

12

1ln)( )6

1ln)( )5

3

22

x

xxxf

xxf

1 ,

23

1 )8

2 ,1

2 )7

2

2

2

3

xx

xxy

xx

xy

32ln of extrema relative theFind )10

cosln )9

2

xxy

θf(x)

Log Rule for Integration

Cuduu

Cxdxx

ln1

ln1

Practice: Don’t forget to use u-substitution when needed.

dx

x

dxx

14

1 )2

2 )1

xdxx

x

dxx

x

3

2

2

13 )4

1 )3

dxxx

x

dxx

x

2

1 )6

tan

sec )5

2

2

dxx

xx

dxx

1

1 )8

23

1 )7

2

2

xdxx

dxx

x

ln

1 )10

1

2 )9 2

xdx

xdx

sec )12

tan )11

Integrals of the Trig Functions

**cotcsclnsec

tanseclnsec

**sinlncot

coslntan

sincos

cossin

Cuuudu

Cuuudu

Cuudu

Cuudu

Cuudu

Cuudu

** Prove these!

40,on tan)( of valueaverage theFind )14

tan1 )13 2

xxf

dxx