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
x
1
ln x   dt, x  0
t
1

y


y





x






x






Definition of e
e
1
ln e   dt  1
t
1
e  2.7182818284
6
Review: Properties of Logs
(i.e. Making life easier!)
ln(1)  0
ln(ab)  ln a  ln b
a
ln   ln a  ln b
b
n
ln a  n ln a
 
Practice: Expand each expression
10
ln 
9
ln 3 x  2 
6x
ln

5
ln
x
x
3
2
3

2
x 1
2

Derivatives of the Natural Log Function
d
1
ln x   ,
dx
x
x0
d
1 du u '
ln u     ,
dx
u dx u
d
1 du u '
ln u    
dx
u dx u
u0
Practice: Find f ’
(Don’t forget your chain rule!!)
1) f ( x)  ln(2 x)
2)
f ( x)  ln(x 2  1)
3)
f ( x)  x ln x
4)
f ( x)  ln x 
3
5)
6)
f ( x)  ln x  1
f ( x)  ln


x x 1
2
2
2 x3 1
7)
8)
3

x  2
y
,
x 1
2
x x2 1
y
,
2
3x  2
x2
x 1
9)
10)
f(x)  ln cosθ


Find therelativeextremaof y  ln x 2  2 x  3
Log Rule for Integration
1
 x dx  ln x  C
1
 u du  ln u  C
Practice:
Don’t forget to use u-substitution when needed.
2
1)  dx
x
2)
1
 4 x  1 dx
3)
x
 x 2  1 dx
4)
3x 2  1
 x3  x d x
5)
sec2 x
 tan x dx
6)
x 1
 x 2  2 x dx
1
7) 
dx
3x  2
8)
x2  x 1
 x 2  1 dx
2x
9) 
dx
2
x  1
10)
1
 x ln x d x
11)
 tan xdx
12)
 sec xdx
Integrals of the Trig Functions
 sin udu   cosu  C
 cosudu  sin u  C
 tanudu   ln cosu  C
 cotudu  ln sin u  C * *
 sec udu  ln sec u  tanu  C
 sec udu   ln csc u  cotu  C * *
** Prove these!
13)

14)
 
Find theaverage value of f ( x)  tan x on 0, 
 4
1  tan2 x dx