Lesson 13: Exponential and Logarithmic Functions (slides)
1. Sec on 3.1–3.2
Exponen al and Logarithmic
Func ons
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
March 9, 2011
.
2. Announcements
Midterm is graded.
average = 44, median=46,
SD =10
There is WebAssign due
a er Spring Break.
Quiz 3 on 2.6, 2.8, 3.1, 3.2
on March 30
3. Midterm Statistics
Average: 43.86/60 = 73.1%
Median: 46/60 = 76.67%
Standard Devia on: 10.64%
“good” is anything above average and “great” is anything more
than one standard devia on above average.
More than one SD below the mean is cause for concern.
4. Objectives for Sections 3.1 and 3.2
Know the defini on of an
exponen al func on
Know the proper es of
exponen al func ons
Understand and apply
the laws of logarithms,
including the change of
base formula.
5. Outline
Defini on of exponen al func ons
Proper es of exponen al Func ons
The number e and the natural exponen al func on
Compound Interest
The number e
A limit
Logarithmic Func ons
6. Derivation of exponentials
Defini on
If a is a real number and n is a posi ve whole number, then
an = a · a · · · · · a
n factors
7. Derivation of exponentials
Defini on
If a is a real number and n is a posi ve whole number, then
an = a · a · · · · · a
n factors
Examples
23 = 2 · 2 · 2 = 8
34 = 3 · 3 · 3 · 3 = 81
(−1)5 = (−1)(−1)(−1)(−1)(−1) = −1
8. Anatomy of a power
Defini on
A power is an expression of the form ab .
The number a is called the base.
The number b is called the exponent.
9. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y
a
(ax )y = axy
(ab)x = ax bx
whenever all exponents are posi ve whole numbers.
10. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y (differences to quo ents)
a
(ax )y = axy
(ab)x = ax bx
whenever all exponents are posi ve whole numbers.
11. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y (differences to quo ents)
a
(ax )y = axy (repeated exponen a on to mul plied powers)
(ab)x = ax bx
whenever all exponents are posi ve whole numbers.
12. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y (differences to quo ents)
a
(ax )y = axy (repeated exponen a on to mul plied powers)
(ab)x = ax bx (power of product is product of powers)
whenever all exponents are posi ve whole numbers.
13. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y (differences to quo ents)
a
(ax )y = axy (repeated exponen a on to mul plied powers)
(ab)x = ax bx (power of product is product of powers)
whenever all exponents are posi ve whole numbers.
14. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
x−y ax
a = y (differences to quo ents)
a
(ax )y = axy (repeated exponen a on to mul plied powers)
(ab)x = ax bx (power of product is product of powers)
whenever all exponents are posi ve whole numbers.
Proof.
Check for yourself:
ax+y = a · a · · · · · a = a · a · · · · · a · a · a · · · · · a = ax ay
x + y factors x factors y factors
15. Let’s be conventional
The desire that these proper es remain true gives us
conven ons for ax when x is not a posi ve whole number.
16. Let’s be conventional
The desire that these proper es remain true gives us
conven ons for ax when x is not a posi ve whole number.
For example, what should a0 be?
We would want this to be true:
!
an = an+0 = an · a0
17. Let’s be conventional
The desire that these proper es remain true gives us
conven ons for ax when x is not a posi ve whole number.
For example, what should a0 be?
We would want this to be true:
n
! ! a
an = an+0 = an · a0 =⇒ a0 = n = 1
a
18. Let’s be conventional
The desire that these proper es remain true gives us
conven ons for ax when x is not a posi ve whole number.
For example, what should a0 be?
We would want this to be true:
n
! ! a
an = an+0 = an · a0 =⇒ a0 = n = 1
a
Defini on
If a ̸= 0, we define a0 = 1.
19. Let’s be conventional
The desire that these proper es remain true gives us
conven ons for ax when x is not a posi ve whole number.
For example, what should a0 be?
We would want this to be true:
n
! ! a
an = an+0 = an · a0 =⇒ a0 = n = 1
a
Defini on
If a ̸= 0, we define a0 = 1.
No ce 00 remains undefined (as a limit form, it’s
indeterminate).
21. Conventions for negative exponents
If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
= n
an a
22. Conventions for negative exponents
If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
= n
an a
Defini on
1
If n is a posi ve integer, we define a−n = .
an
23. Defini on
1
If n is a posi ve integer, we define a−n = .
an
24. Defini on
1
If n is a posi ve integer, we define a−n = .
an
Fact
1
The conven on that a−n = “works” for nega ve n as well.
an
m−n am
If m and n are any integers, then a = n.
a
26. Conventions for fractional exponents
If q is a posi ve integer, we want
! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a
27. Conventions for fractional exponents
If q is a posi ve integer, we want
! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a
Defini on
√
If q is a posi ve integer, we define a1/q = q
a. We must have a ≥ 0
if q is even.
28. Conventions for fractional exponents
If q is a posi ve integer, we want
! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a
Defini on
√
If q is a posi ve integer, we define a1/q = q a. We must have a ≥ 0
if q is even.
√q
(√ )p
No ce that ap = q a . So we can unambiguously say
ap/q = (ap )1/q = (a1/q )p
30. Conventions for irrational
exponents
So ax is well-defined if a is posi ve and x is ra onal.
What about irra onal powers?
Defini on
Let a > 0. Then
ax = lim ar
r→x
r ra onal
31. Conventions for irrational
exponents
So ax is well-defined if a is posi ve and x is ra onal.
What about irra onal powers?
Defini on
Let a > 0. Then
ax = lim ar
r→x
r ra onal
In other words, to approximate ax for irra onal x, take r close to x
but ra onal and compute ar .
32. Approximating a power with an
irrational exponent
r 2r
3 23
√=8
10
3.1 231/10 = √ 31 ≈ 8.57419
2
100
3.14 2314/100 = √ 314 ≈ 8.81524
2
1000
3.141 23141/1000 = 23141 ≈ 8.82135
The limit (numerically approximated is)
2π ≈ 8.82498
39. Graphs of exponential functions
y
y = (1/2)x y = 10x 3x = 2x
y= y y = 1.5x
y = 1x
. x
40. Graphs of exponential functions
y
y = (y/= x(1/3)x
1 2) y = 10x 3x = 2x
y= y y = 1.5x
y = 1x
. x
41. Graphs of exponential functions
y
y = (y/= x(1/3)x
1 2) y = (1/10y x= 10x 3x = 2x
) y= y y = 1.5x
y = 1x
. x
42. Graphs of exponential functions
y
y y =y/=3(1/3)x
= (1(2/x)x
2) y = (1/10y x= 10x 3x = 2x
) y= y y = 1.5x
y = 1x
. x
43. Outline
Defini on of exponen al func ons
Proper es of exponen al Func ons
The number e and the natural exponen al func on
Compound Interest
The number e
A limit
Logarithmic Func ons
44. Properties of exponential Functions
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with
domain (−∞, ∞) and range (0, ∞). In par cular, ax > 0 for all x.
For any real numbers x and y, and posi ve numbers a and b we have
ax+y = ax ay
x−y ax
a = y
a
(a ) = axy
x y
(ab)x = ax bx
45. Properties of exponential Functions
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with
domain (−∞, ∞) and range (0, ∞). In par cular, ax > 0 for all x.
For any real numbers x and y, and posi ve numbers a and b we have
ax+y = ax ay
x−y ax
a = y (nega ve exponents mean reciprocals)
a
(a ) = axy
x y
(ab)x = ax bx
46. Properties of exponential Functions
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with
domain (−∞, ∞) and range (0, ∞). In par cular, ax > 0 for all x.
For any real numbers x and y, and posi ve numbers a and b we have
ax+y = ax ay
x−y ax
a = y (nega ve exponents mean reciprocals)
a
(a ) = axy (frac onal exponents mean roots)
x y
(ab)x = ax bx
47. Proof.
This is true for posi ve integer exponents by natural defini on
Our conven onal defini ons make these true for ra onal
exponents
Our limit defini on make these for irra onal exponents, too
53. Limits of exponential functions
Fact (Limits of exponen al
func ons) y
y (1 y )/3 x
y = =/(1= )(2/3)x y = y1/1010= 2x = 1.5
2
x
( = =x 3x y
y ) x
y
If a > 1, then
lim ax = ∞ and
x→∞
lim ax = 0
x→−∞
If 0 < a < 1, then y = 1x
lim ax = 0 and . x
x→∞
lim ax = ∞
x→−∞
54. Outline
Defini on of exponen al func ons
Proper es of exponen al Func ons
The number e and the natural exponen al func on
Compound Interest
The number e
A limit
Logarithmic Func ons
55. Compounded Interest
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have A er one year?
A er two years? A er t years?
56. Compounded Interest
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have A er one year?
A er two years? A er t years?
Answer
$100 + 10% = $110
57. Compounded Interest
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have A er one year?
A er two years? A er t years?
Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
58. Compounded Interest
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have A er one year?
A er two years? A er t years?
Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
$100(1.1)t .
59. Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
60. Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
Answer
$100(1.025)4 = $110.38,
61. Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
62. Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
$100(1.025)8 = $121.84
63. Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
$100(1.025)8 = $121.84
$100(1.025)4t .
64. Compounded Interest: monthly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded twelve mes a year. How much do you have a er t
years?
65. Compounded Interest: monthly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded twelve mes a year. How much do you have a er t
years?
Answer
$100(1 + 10%/12)12t
66. Compounded Interest: general
Ques on
Suppose you save P at interest rate r, with interest compounded n
mes a year. How much do you have a er t years?
67. Compounded Interest: general
Ques on
Suppose you save P at interest rate r, with interest compounded n
mes a year. How much do you have a er t years?
Answer
( r )nt
B(t) = P 1 +
n
68. Compounded Interest: continuous
Ques on
Suppose you save P at interest rate r, with interest compounded
every instant. How much do you have a er t years?
69. Compounded Interest: continuous
Ques on
Suppose you save P at interest rate r, with interest compounded
every instant. How much do you have a er t years?
Answer
( ( )rnt
r )nt 1
B(t) = lim P 1 + = lim P 1 +
n→∞ n n→∞ n
[ ( )n ]rt
1
= P lim 1 +
n→∞ n
independent of P, r, or t
74. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
75. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
76. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
77. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106 2.71828
78. Existence of e
See Appendix B
( )n
1
n 1+
We can experimentally n
verify that this number 1 2
exists and is 2 2.25
3 2.37037
e ≈ 2.718281828459045 . . .
10 2.59374
100 2.70481
1000 2.71692
106 2.71828
79. Existence of e
See Appendix B
( )n
1
n 1+
We can experimentally n
verify that this number 1 2
exists and is 2 2.25
3 2.37037
e ≈ 2.718281828459045 . . .
10 2.59374
e is irra onal 100 2.70481
1000 2.71692
106 2.71828
80. Existence of e
See Appendix B
( )n
1
n 1+
We can experimentally n
verify that this number 1 2
exists and is 2 2.25
3 2.37037
e ≈ 2.718281828459045 . . .
10 2.59374
e is irra onal 100 2.70481
1000 2.71692
e is transcendental
106 2.71828
81. Meet the Mathematician: Leonhard Euler
Born in Switzerland, lived
in Prussia (Germany) and
Russia
Eyesight trouble all his
life, blind from 1766
onward
Hundreds of
contribu ons to calculus,
number theory, graph
theory, fluid mechanics, Leonhard Paul Euler
op cs, and astronomy Swiss, 1707–1783
83. A limit
Ques on
eh − 1
What is lim ?
h→0 h
Answer
e = lim (1 + 1/n)n = lim (1 + h)1/h . So for a small h,
n→∞ h→0
e ≈ (1 + h) 1/h
. So
[ ]h
eh − 1 (1 + h)1/h − 1
≈ =1
h h
84. A limit
eh − 1
It follows that lim = 1.
h→0 h
2h − 1
This can be used to characterize e: lim = 0.693 · · · < 1
h→0 h
3h − 1
and lim = 1.099 · · · > 1
h→0 h
85. Outline
Defini on of exponen al func ons
Proper es of exponen al Func ons
The number e and the natural exponen al func on
Compound Interest
The number e
A limit
Logarithmic Func ons
86. Logarithms
Defini on
The base a logarithm loga x is the inverse of the func on ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex . So
y = ln x ⇐⇒ x = ey .
96. Graphs of logarithmic functions
y
y =y3= 2x
x
y = log2 x
y = log3 x
(0, 1)
.
(1, 0) x
97. Graphs of logarithmic functions
y
y =y10y3= 2x
=x x
y = log2 x
y = log3 x
(0, 1)
y = log10 x
.
(1, 0) x
98. Graphs of logarithmic functions
y
y =x ex
y =y10y3= 2x
= x
y = log2 x
yy= log3 x
= ln x
(0, 1)
y = log10 x
.
(1, 0) x
99. Change of base formula for logarithms
Fact
logb x
If a > 0 and a ̸= 1, and the same for b, then loga x =
logb a
100. Change of base formula for logarithms
Fact
logb x
If a > 0 and a ̸= 1, and the same for b, then loga x =
logb a
Proof.
If y = loga x, then x = ay
So logb x = logb (ay ) = y logb a
Therefore
logb x
y = loga x =
logb a
102. Example of changing base
Example
Find log2 8 by using log10 only.
Solu on
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
103. Example of changing base
Example
Find log2 8 by using log10 only.
Solu on
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised?
104. Example of changing base
Example
Find log2 8 by using log10 only.
Solu on
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised? No, log2 8 = log2 23 = 3 directly.
105. Upshot of changing base
The point of the change of base formula
logb x 1
loga x = = · logb x = constant · logb x
logb a logb a
is that all the logarithmic func ons are mul ples of each other. So
just pick one and call it your favorite.
Engineers like the common logarithm log = log10
Computer scien sts like the binary logarithm lg = log2
Mathema cians like natural logarithm ln = loge
Naturally, we will follow the mathema cians. Just don’t pronounce
it “lawn.”