NEWlogarithm - The
natural
logarithm
func1on
...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: The
natural
logarithm
func1on
 Sec1on
7.2*
 The
natural
logarithm
func1on
 •  •  •  •  •  •  Defini1on
 Graph
 Proper1es
 Deriva1ves
that
involve
logarithm

 Integrals
that
involve
logarithm
 Logarithm
in
different
bases
 in
base
e=2.718281828…
 The
natural
logarithm
func1on
 •  •  •  •  •  •  Defini1on
 Graph
 Proper1es
 Deriva1ves
that
involve
logarithm

 Integrals
that
involve
logarithm
 Logarithm
in
different
bases
 Defini1on
of
natural
logarithm
 For
x>0,
we
define
 Fix
x>0.
The
func1on
f(t)=1/t

 is
con1nuous
over
[1,x],
so
 this
integral
exists.
 Natural
logarithm
in
terms
of
area
 f
con1nuous
over
[a,b],
and
posi1ve
 [a,b]
is
an
interval
 >a
 b ∫ a f ( t ) dt = area beneath the graph of f , over [ a, b] € Natural
logarithm
in
terms
of
area
 con1nuous
over
[1,u],
and
posi1ve
 for
u>1,
[1,u]
is
an
interval



 >1
 Fix
u>1.
 1 A( u) = ∫ dt 1t u That’s
what
we
defined
as
ln(u)
 Natural
logarithm
in
terms
of
area
 1 ln( x ) = ∫ dt = A( x ) for x ≥ 1 1t x € 1 1 ln( x ) = ∫ dt = − ∫ dt = − A( x ) for 0 < x < 1 1t xt x 1 € Sign
of
the
Natural
logarithm
 ln( x ) = A( x ) for x ≥ 1 
 € ln(x)>0
for
x>1,
and
 ln(x)=0
for
x=1

 ln( x ) = − A( x ) for 0 < x < 1 
 ln(x)<0
for
0<x<1
 € 1 The
natural
logarithm
ln( x ) = ∫ dt 1t in
base
e=2.718281828…
 x € Ques1on
 The
shaded
area
is
equal
to:
 A)
‐1/e2
+1
 B)
1/e
 C)
1
 The
natural
logarithm
func1on
 •  •  •  •  •  •  Defini1on

✔
 Graph
 Proper1es
 Deriva1ves
that
involve
logarithm

 Integrals
that
involve
logarithm
 Logarithm
in
different
bases
 Graph
of
the
func1on
f(x)=ln(x)
 • Domain:
x>0
 already
 know:
 0
 1
 x
 • x‐intercept:
x=1
 • Posi1ve:
x>1
 • Nega1ve:
0<x<1
 You
need
more
informa2on:
 • 
behavior
at
end‐points
of
domain


 • 
deriva2ves
(extremes
etc..)

 Graph
of
the
func1on
f(x)=ln(x)
 Behavior
at
end‐point
of
domain:
 x lim ln( x ) = lim ∫ x →∞ x →∞ 1 x 1 dt = +∞ t 1 lim ln( x ) = lim ∫ dt = −∞ x →0 x →0 1t 0
 1
 x
 Graph
of
the
func1on
f(x)=ln(x)
 DerivaAve:
 d d ln( x ) = dx dx 1 1 ∫ t dt = x 1 x d ln( x ) > 0 for all x > 0 dx Fundamental
theorem
of
calculus
 € The
func4on
ln(x)
is
increasing
for
x>0.
 It
has
no
maxima
and
no
minima.
 Graph
of
the
func1on
f(x)=ln(x)
 The
natural
logarithm
func1on
 •  •  •  •  •  •  Defini1on

✔
 Graph

✔
 Proper1es
 Deriva1ves
that
involve
logarithm

 Integrals
that
involve
logarithm
 Logarithm
in
different
bases
 Proper1es
of
the
logarithm



 1)

ln(ab)=ln(a)+ln(b)
 2)

ln(a/b)=ln(a)‐ln(b)
 3)

ln(ar)
=
r
ln(a),
for
all
r
ra1onal
 Proper1es
of
the
logarithm



 1)

ln(ab)=ln(a)+ln(b)
 By
subs1tu1on:
t=x/a
 1/x





1/(at)
 dx







a
dt
 a









1














 ab







b

 Proof:
 
 ln(a)
 
 Proper1es
of
the
logarithm



 2)

ln(a/b)=ln(a)‐ln(b)
 Proof:
 0
=
ln(1)
=
ln(b/b)
=
ln(b(1/b))
=
ln(b)
+
ln(1/b)
 
ln(1/b)
=
‐
ln(b)

 
ln(a/b)
=
ln(a(1/b))
=
ln(a)
+
ln(1/b)
=
ln(a)
–
ln(b)

 
 
 
 Proper1es
of
the
logarithm



 3)

ln(ar)
=
r
ln(a),
for
all
r
ra1onal
 E.g.,
 ln(a3)=ln(a3/2a3/2)=ln(a3/2)+ln(a3/2)
=
2
ln(a3/2)





ln(a3/2)=(1/2)
ln(a3)




 ln(a3)
=
ln(aaa)=
ln(a)
+
ln(a)
+
ln(a)
=
3
ln(a)
 
ln(a3/2)=(1/2)
ln(a3)
=
(1/2)
3
ln(a)
=
(1/2)
3
ln(a)






 Proper1es
of
the
logarithm



 1)

ln(ab)=ln(a)+ln(b)
 2)

ln(a/b)=ln(a)‐ln(b)
 3)

ln(ar)
=
r
ln(a),
for
all
r
ra1onal
 Example:
 ln ( x 2 + 4 ) 5 sin x x +1 2 3 = ln[( x 2 + 4 ) 5 sin x ] − ln [ x3 +1 = 1 = 5 ln( x + 4 ) + ln(sin x ) − ln( x 3 + 1) 2 Ques1on
 The
quan1ty


























is
equal
to
 ln a(b 2 + c 2 ) ln( a) ln(b 2 + c 2 ) A) + €2 2 B) − 2 ln( a) − 2 ln(b + c ) ln( a) ln(2b + 2c ) + 2 2 2 2 C) The
natural
logarithm
func1on
 •  •  •  •  •  •  Defini1on

✔
 Graph

✔
 Proper1es
✔
 Deriva1ves
that
involve
logarithm

 Integrals
that
involve
logarithm
 Logarithm
in
different
bases
 Deriva1ves
of
logarithm
func1on
 d 1 ln( f ( x )) = f '( x) dx f ( x) € ⏎
 d d ln( x ) = dx dx 1 1 ∫ t dt = x 1 x Chain
rule
 Ques1on
 The
deriva1ve
of
the
func1on
f(x)=ln(tan(x))
is
 A) 1 tan x B) 1 sec 2 x tan x 2 1 C ) sec x Ques1on
 The
deriva1ve
of
the
func1on

f(x)=ln(x2sin2x)

is
 A) 1 x 2 sin 2 x 1 1 +2 2 x sin x 2 2 cos x + x sin x B) C) Deriva1ves
of
logarithm
func1on
 d 1 ln( f ( x )) = f '( x) dx f ( x) € Hint:
To
simplify
the
calcula2ons,
use
the
proper2es
of
the
logarithm
BEFORE
differen2a2ng.
 1)

ln(ab)=ln(a)+ln(b)
 2)

ln(a/b)=ln(a)‐ln(b)
 3)

ln(ar)
=
r
ln(a),
for
all
r
ra1onal
 • 
Use
the
formula
directly:

 d 1 ln( f ( x )) = f '( x) dx f ( x) d ( x 2 + 4 ) 5 sin x 1 d ( x 2 + 4 ) 5 sin x ln =2 3 3 ( x + 4 ) 5 sin x dx dx x +1 x +1 € 3 x +1 f(x)
 1/f(x)
 f’(x)
 RATHER
 COMPLICATED!
 • 
Simplify
first:

 ( x 2 + 4 ) 5 sin x d 1 2 3 ln = 5 ln( x + 4 ) + ln(sin x ) − 2 ln( x + 1) 3 x + 1 dx 1 1 11 MUCH
EASIER!
 =5 2 (2 x ) + (cos x ) − (3 x 2 ) 3 x +4 sin x 2 ( x + 1) d dx The
natural
logarithm
func1on
 •  •  •  •  •  •  Defini1on

✔
 Graph

✔
 Proper1es
✔
 Deriva1ves
that
involve
logarithm
✔
 Integrals
that
involve
logarithm
 Logarithms
in
different
bases
 Logarithms
and
Integra1on
 x if x > 0 | x |= − x if x > 0 ln( x ) if x > 0 ⇒ ln | x | = ln(− x ) if x > 0 d 1 if x > 0 dx ln( x ) if x > 0 x d ⇒ ln | x | = = = d 1 dx ln(− x ) if x > 0 (−1) if x > 0 dx −x 1 x ∫ 1 dx = ln | x | + C x 
 Logarithms
and
Integra1on
 ∫ 1 dx = ln | x | + C x Then
the
subs1tu1on
rule
gives:
 € 1 1 ∫ f ( x ) f ' ( x ) dx = ∫ u du = ln | u | + C = ln | f ( x ) | + C u=f(x)
 Example:
 ∫ tan x dx = ∫ 1 cos x dx = sin x ∫ 1 du = ln | u | + C = ln | sin x | + C u Ques1on
 (ln x ) 2 The
integral





















is
solved
with
the
subs1tu1on
 ∫ x dx € A) u = 1 / x B) u = ln x C ) u = (ln x ) 2 The
natural
logarithm
func1on
 •  •  •  •  •  •  Defini1on

✔
 Graph

✔
 Proper1es
✔
 Deriva1ves
that
involve
logarithm
✔
 Integrals
that
involve
logarithm
✔
 Logarithms
in
different
bases
 Change
of
base
law
 •  The
natural
logarithm
ln(x)
means
log
in
base
e
 •  e
is
an
irra1onal
number
≈
2.718281828
 log e x ln x •  If
a≠1
and
a>0,
 log a x = = log e a ln a Deriva1ve
of
y=logax
 1/x
 ...
View Full Document

This note was uploaded on 12/07/2010 for the course MATH MATH 2B taught by Professor Famiglietti,c during the Fall '09 term at UC Irvine.

Ask a homework question - tutors are online