1.
Sections 3.1–3.2
Exponential and Logarithmic Functions
V63.0121.041, Calculus I
New York University
October 20, 2010
Announcements
Midterm is graded and scores are on blackboard. Should get it
back in recitation.
There is WebAssign due Monday/Tuesday next week.
. . . . . .
2.
Announcements
Midterm is graded and
scores are on blackboard.
Should get it back in
recitation.
There is WebAssign due
Monday/Tuesday next
week.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 2 / 37
3.
Objectives for Sections 3.1 and 3.2
Know the definition of an
exponential function
Know the properties of
exponential functions
Understand and apply the
laws of logarithms,
including the change of
base formula.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 3 / 37
4.
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
Logarithmic Functions
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 4 / 37
5.
Derivation of exponential functions
Definition
If a is a real number and n is a positive whole number, then
an = a · a · · · · · a
n factors
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 5 / 37
6.
Derivation of exponential functions
Definition
If a is a real number and n is a positive 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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 5 / 37
7.
Anatomy of a power
Definition
A power is an expression of the form ab .
The number a is called the base.
The number b is called the exponent.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 6 / 37
8.
Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y
a
(ax )y = axy
(ab)x = ax bx
whenever all exponents are positive whole numbers.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
9.
Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quotients)
a
(ax )y = axy
(ab)x = ax bx
whenever all exponents are positive whole numbers.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
10.
Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quotients)
a
(ax )y = axy (repeated exponentiation to multiplied powers)
(ab)x = ax bx
whenever all exponents are positive whole numbers.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
11.
Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quotients)
a
(ax )y = axy (repeated exponentiation to multiplied powers)
(ab)x = ax bx (power of product is product of powers)
whenever all exponents are positive whole numbers.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
12.
Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quotients)
a
(ax )y = axy (repeated exponentiation to multiplied powers)
(ab)x = ax bx (power of product is product of powers)
whenever all exponents are positive whole numbers.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
13.
Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quotients)
a
(ax )y = axy (repeated exponentiation to multiplied powers)
(ab)x = ax bx (power of product is product of powers)
whenever all exponents are positive 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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
14.
Let's be conventional
The desire that these properties remain true gives us conventions
for ax when x is not a positive whole number.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
15.
Let's be conventional
The desire that these properties remain true gives us conventions
for ax when x is not a positive whole number.
For example, what should a0 be? We cannot write down zero a’s
and multiply them together. But we would want this to be true:
!
an = an+0 = an · a0
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
16.
Let's be conventional
The desire that these properties remain true gives us conventions
for ax when x is not a positive whole number.
For example, what should a0 be? We cannot write down zero a’s
and multiply them together. But we would want this to be true:
! ! an
an = an+0 = an · a0 =⇒ a0 = =1
an
(The equality with the exclamation point is what we want.)
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
17.
Let's be conventional
The desire that these properties remain true gives us conventions
for ax when x is not a positive whole number.
For example, what should a0 be? We cannot write down zero a’s
and multiply them together. But we would want this to be true:
! ! an
an = an+0 = an · a0 =⇒ a0 = =1
an
(The equality with the exclamation point is what we want.)
Definition
If a ̸= 0, we define a0 = 1.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
18.
Let's be conventional
The desire that these properties remain true gives us conventions
for ax when x is not a positive whole number.
For example, what should a0 be? We cannot write down zero a’s
and multiply them together. But we would want this to be true:
! ! an
an = an+0 = an · a0 =⇒ a0 = =1
an
(The equality with the exclamation point is what we want.)
Definition
If a ̸= 0, we define a0 = 1.
Notice 00 remains undefined (as a limit form, it’s indeterminate).
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
19.
Conventions for negative exponents
If n ≥ 0, we want
an+(−n) = an · a−n
!
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
20.
Conventions for negative exponents
If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
n
= n
a a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
21.
Conventions for negative exponents
If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
n
= n
a a
Definition
1
If n is a positive integer, we define a−n = .
an
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
22.
Conventions for negative exponents
If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
n
= n
a a
Definition
1
If n is a positive integer, we define a−n = .
an
Fact
1
The convention that a−n = “works” for negative n as well.
an
am
If m and n are any integers, then am−n = n .
a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
23.
Conventions for fractional exponents
If q is a positive integer, we want
!
(a1/q )q = a1 = a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
24.
Conventions for fractional exponents
If q is a positive integer, we want
! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
25.
Conventions for fractional exponents
If q is a positive integer, we want
! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a
Definition
√
If q is a positive integer, we define a1/q = q
a. We must have a ≥ 0 if q
is even.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
26.
Conventions for fractional exponents
If q is a positive integer, we want
! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a
Definition
√
If q is a positive integer, we define a1/q = q a. We must have a ≥ 0 if q
is even.
√q
( √ )p
Notice that ap = q a . So we can unambiguously say
ap/q = (ap )1/q = (a1/q )p
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
27.
Conventions for irrational exponents
So ax is well-defined if a is positive and x is rational.
What about irrational powers?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 37
28.
Conventions for irrational exponents
So ax is well-defined if a is positive and x is rational.
What about irrational powers?
Definition
Let a > 0. Then
ax = lim ar
r→x
r rational
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 37
29.
Conventions for irrational exponents
So ax is well-defined if a is positive and x is rational.
What about irrational powers?
Definition
Let a > 0. Then
ax = lim ar
r→x
r rational
In other words, to approximate ax for irrational x, take r close to x but
rational and compute ar .
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 37
30.
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 = √2314 ≈ 8.81524
1000
3.141 23141/1000 = 23141 ≈ 8.82135
The limit (numerically approximated is)
2π ≈ 8.82498
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 12 / 37
31.
Graphs of various exponential functions
y
.
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
32.
Graphs of various exponential functions
y
.
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
33.
Graphs of various exponential functions
y
.
. = 2x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
34.
Graphs of various exponential functions
y
.
. = 3x. = 2x
y y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
35.
Graphs of various exponential functions
y
.
. = 10x= 3x. = 2x
y y
. y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
36.
Graphs of various exponential functions
y
.
. = 10x= 3x. = 2x
y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
37.
Graphs of various exponential functions
y
.
. = (1/2)x
y . = 10x= 3x. = 2x
y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
38.
Graphs of various exponential functions
x
y
.
. = (1/2)x (1/3)
y y
. = . = 10x= 3x. = 2x
y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
39.
Graphs of various exponential functions
y
.
. = (1/2)x (1/3)
y y
. = x
. = (1/10)x. = 10x= 3x. = 2x
y y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
40.
Graphs of various exponential functions
y
.
y yx
.. = ((1/2)x (1/3)x
y = 2/. )=
3 . = (1/10)x. = 10x= 3x. = 2x
y y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
41.
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
Logarithmic Functions
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 14 / 37
42.
Properties of exponential Functions
.
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain
(−∞, ∞) and range (0, ∞). In particular, ax > 0 for all x. For any real numbers
x and y, and positive numbers a and b we have
ax+y = ax ay
ax
ax−y = y
a
(ax )y = axy
(ab)x = ax bx
Proof.
This is true for positive integer exponents by natural definition
Our conventional definitions make these true for rational exponents
Our limit definition make these for irrational exponents, too
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 15 / 37
43.
Properties of exponential Functions
.
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain
(−∞, ∞) and range (0, ∞). In particular, ax > 0 for all x. For any real numbers
x and y, and positive numbers a and b we have
ax+y = ax ay
ax
ax−y = y (negative exponents mean reciprocals)
a
(ax )y = axy
(ab)x = ax bx
Proof.
This is true for positive integer exponents by natural definition
Our conventional definitions make these true for rational exponents
Our limit definition make these for irrational exponents, too
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 15 / 37
44.
Properties of exponential Functions
.
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain
(−∞, ∞) and range (0, ∞). In particular, ax > 0 for all x. For any real numbers
x and y, and positive numbers a and b we have
ax+y = ax ay
ax
ax−y = y (negative exponents mean reciprocals)
a
(ax )y = axy (fractional exponents mean roots)
(ab)x = ax bx
Proof.
This is true for positive integer exponents by natural definition
Our conventional definitions make these true for rational exponents
Our limit definition make these for irrational exponents, too
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 15 / 37
45.
Simplifying exponential expressions
Example
Simplify: 82/3
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 16 / 37
50.
Limits of exponential functions
Fact (Limits of exponential y
.
functions) . = (= 2()1/32/3)x
y . 1/ =x( )x
y .
y y y = x . 3x y
. = (. /10)10x= 2x. =
1 . =
y y
If a > 1, then lim ax = ∞
x→∞
and lim ax = 0
x→−∞
If 0 < a < 1, then
lim ax = 0 and y
. =
x→∞
lim a = ∞ x . x
.
x→−∞
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 17 / 37
51.
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
Logarithmic Functions
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 18 / 37
52.
Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
53.
Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100 + 10% = $110
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
54.
Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
55.
Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
$100(1.1)t .
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
56.
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
57.
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4 = $110.38,
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
58.
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
59.
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
$100(1.025)8 = $121.84
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
60.
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
$100(1.025)8 = $121.84
$100(1.025)4t .
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
61.
Compounded Interest: monthly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded twelve times a year. How much do you have after t
years?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 21 / 37
62.
Compounded Interest: monthly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded twelve times a year. How much do you have after t
years?
Answer
$100(1 + 10%/12)12t
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 21 / 37
63.
Compounded Interest: general
Question
Suppose you save P at interest rate r, with interest compounded n
times a year. How much do you have after t years?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 22 / 37
64.
Compounded Interest: general
Question
Suppose you save P at interest rate r, with interest compounded n
times a year. How much do you have after t years?
Answer
( r )nt
B(t) = P 1 +
n
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 22 / 37
65.
Compounded Interest: continuous
Question
Suppose you save P at interest rate r, with interest compounded every
instant. How much do you have after t years?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 23 / 37
66.
Compounded Interest: continuous
Question
Suppose you save P at interest rate r, with interest compounded every
instant. How much do you have after t years?
Answer
( ( )
r )nt 1 rnt
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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 23 / 37
67.
The magic number
Definition
( )
1 n
e = lim 1 +
n→∞ n
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 24 / 37
68.
The magic number
Definition
( )
1 n
e = lim 1 +
n→∞ n
So now continuously-compounded interest can be expressed as
B(t) = Pert .
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 24 / 37
69.
Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
70.
Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
3 2.37037
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
71.
Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
72.
Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
73.
Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
74.
Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106 2.71828
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
75.
Existence of e
See Appendix B
( )
1 n
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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
76.
Existence of e
See Appendix B
( )
1 n
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 irrational 100 2.70481
1000 2.71692
106 2.71828
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
77.
Existence of e
See Appendix B
( )
1 n
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 irrational 100 2.70481
1000 2.71692
e is transcendental
106 2.71828
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
78.
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 contributions
to calculus, number theory,
graph theory, fluid
mechanics, optics, and
astronomy
Leonhard Paul Euler
Swiss, 1707–1783
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 26 / 37
79.
A limit
.
Question
eh − 1
What is lim ?
h→0 h
. . . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 37
80.
A limit
.
Question
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, e ≈ (1 + h)1/h .
n h→0
So [ ]h
eh − 1 (1 + h)1/h − 1
≈ =1
h h
. . . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 37
81.
A limit
.
Question
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, e ≈ (1 + h)1/h .
n h→0
So [ ]h
eh − 1 (1 + h)1/h − 1
≈ =1
h h
eh − 1
It follows that lim = 1.
h→0 h
2h − 1
This can be used to characterize e: lim = 0.693 · · · < 1 and
h→0 h
3h − 1
lim = 1.099 · · · > 1
h→0 h
. . . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 37
82.
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
Logarithmic Functions
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 28 / 37
83.
Logarithms
Definition
The base a logarithm loga x is the inverse of the function ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex . So
y = ln x ⇐⇒ x = ey .
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
84.
Logarithms
Definition
The base a logarithm loga x is the inverse of the function ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex . So
y = ln x ⇐⇒ x = ey .
Facts
(i) loga (x1 · x2 ) = loga x1 + loga x2
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
85.
Logarithms
Definition
The base a logarithm loga x is the inverse of the function ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex . So
y = ln x ⇐⇒ x = ey .
Facts
(i) loga (x1 · x2 ) = loga x1 + loga x2
( )
x1
(ii) loga = loga x1 − loga x2
x2
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
86.
Logarithms
Definition
The base a logarithm loga x is the inverse of the function ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex . So
y = ln x ⇐⇒ x = ey .
Facts
(i) loga (x1 · x2 ) = loga x1 + loga x2
( )
x1
(ii) loga = loga x1 − loga x2
x2
(iii) loga (xr ) = r loga x
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
87.
Logarithms convert products to sums
Suppose y1 = loga x1 and y2 = loga x2
Then x1 = ay1 and x2 = ay2
So x1 x2 = ay1 ay2 = ay1 +y2
Therefore
loga (x1 · x2 ) = loga x1 + loga x2
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 30 / 37
88.
Example
Write as a single logarithm: 2 ln 4 − ln 3.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
89.
Example
Write as a single logarithm: 2 ln 4 − ln 3.
Solution
42
2 ln 4 − ln 3 = ln 42 − ln 3 = ln
3
ln 42
not !
ln 3
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
90.
Example
Write as a single logarithm: 2 ln 4 − ln 3.
Solution
42
2 ln 4 − ln 3 = ln 42 − ln 3 = ln
3
ln 42
not !
ln 3
Example
3
Write as a single logarithm: ln + 4 ln 2
4
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
91.
Example
Write as a single logarithm: 2 ln 4 − ln 3.
Solution
42
2 ln 4 − ln 3 = ln 42 − ln 3 = ln
3
ln 42
not !
ln 3
Example
3
Write as a single logarithm: ln + 4 ln 2
4
Answer
ln 12
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
92.
Graphs of logarithmic functions
y
.
. = 2x
y
y
. = log2 x
. . 0, 1)
(
..1, 0) .
( x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
93.
Graphs of logarithmic functions
y
.
. = 3x= 2x
y . y
y
. = log2 x
y
. = log3 x
. . 0, 1)
(
..1, 0) .
( x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
94.
Graphs of logarithmic functions
y
.
. = .10x 3x= 2x
y y= . y
y
. = log2 x
y
. = log3 x
. . 0, 1)
(
y
. = log10 x
..1, 0) .
( x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
95.
Graphs of logarithmic functions
y
.
. = .10=3x= 2x
y xy
y y. = .ex
y
. = log2 x
y
. = ln x
y
. = log3 x
. . 0, 1)
(
y
. = log10 x
..1, 0) .
( x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
96.
Change of base formula for exponentials
Fact
If a > 0 and a ̸= 1, then
ln x
loga x =
ln a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 33 / 37
97.
Change of base formula for exponentials
Fact
If a > 0 and a ̸= 1, then
ln x
loga x =
ln a
Proof.
If y = loga x, then x = ay
So ln x = ln(ay ) = y ln a
Therefore
ln x
y = loga x =
ln a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 33 / 37
98.
Example of changing base
Example
Find log2 8 by using log10 only.
Surprised?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 34 / 37
99.
Example of changing base
Example
Find log2 8 by using log10 only.
Solution
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 34 / 37
100.
Example of changing base
Example
Find log2 8 by using log10 only.
Solution
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised? No, log2 8 = log2 23 = 3 directly.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 34 / 37
101.
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 functions are multiples of each other. So just
pick one and call it your favorite.
Engineers like the common logarithm log = log10
Computer scientists like the binary logarithm lg = log2
Mathematicians like natural logarithm ln = loge
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 35 / 37
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