1.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Sections 3.1–3.2 Notes
Exponential and Logarithmic Functions
V63.0121.021, Calculus I
New York University
October 21, 2010
Announcements
Midterm is graded and scores are on blackboard. Should get it back
in recitation.
There is WebAssign due Monday/Tuesday next week.
Announcements
Notes
Midterm is graded and
scores are on blackboard.
Should get it back in
recitation.
There is WebAssign due
Monday/Tuesday next week.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 2 / 38
Midterm Statistics
Notes
Average: 78.77%
Median: 80%
Standard Deviation: 12.39%
“good” is anything above average and “great” is anything more than
one standard deviation above average.
More than one SD below the mean is cause for concern.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 3 / 38
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2.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Objectives for Sections 3.1 and 3.2
Notes
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.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 4 / 38
Outline
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 5 / 38
Derivation of exponential functions
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 6 / 38
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3.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Anatomy of a power
Notes
Deﬁnition
A power is an expression of the form ab .
The number a is called the base.
The number b is called the exponent.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 7 / 38
Fact
Notes
If a is a real number, then
x+y x y
a = a a (sums to products)
x−y ax
a = y (diﬀerences to quotients)
a
(ax )y = axy (repeated exponentiation to multiplied powers)
(ab)x = ax b x (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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 8 / 38
Let’s be conventional
Notes
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).
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 9 / 38
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4.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Conventions for negative exponents
Notes
If n ≥ 0, we want
! ! a0 1
an+(−n) = an · a−n =⇒ a−n = = n
an 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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 10 / 38
Conventions for fractional exponents
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 11 / 38
Conventions for irrational exponents
Notes
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 .
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 12 / 38
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5.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Approximating a power with an irrational exponent
Notes
r 2r
3
3 2 =8
√
10
3.1 231/10 = √ 31 ≈ 8.57419
2
314/100 100
3.14 2 = √ 314 ≈ 8.81524
2
1000
3.141 23141/1000 = 23141 ≈ 8.82135
The limit (numerically approximated is)
2π ≈ 8.82498
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 13 / 38
Graphs of various exponential functions
y Notes
y = ((21/2))xx (1/3)x
y = /3 =y y = (1/10)xy = 10x= 3xy = 2x
y y = 1.5x
y = 1x
x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
Outline
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 15 / 38
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6.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Properties of exponential Functions
Notes
Theorem
x
If a > 0 and a = 1, then f (x) = a 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)
x y
a xy
(a ) = a (fractional exponents mean roots)
(ab)x = ax b x
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 16 / 38
Simplifying exponential expressions
Notes
Example
Simplify: 82/3
Solution
√
3
√
82/3 = 82 = 64 = 4
3
√ 2
8 = 22 = 4.
3
Or,
Example
√
8
Simplify:
21/2
Answer
2
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 17 / 38
Limits of exponential functions
Notes
Fact (Limits of exponential y
y = (= 2(1/(2/3)x y = (y/10)10x3x 2xy = 1.5x
y 1/ = 3)x
y )x 1 = xy =
y=
functions)
If a > 1, then lim ax = ∞
x→∞
and lim ax = 0
x→−∞
If 0 < a < 1, then
lim ax = 0 and y = 1x
x→∞
lim ax = ∞ x
x→−∞
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 18 / 38
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7.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Outline
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 19 / 38
Compounded Interest
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 20 / 38
Compounded Interest: quarterly
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
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8.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Compounded Interest: monthly
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 22 / 38
Compounded Interest: general
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 23 / 38
Compounded Interest: continuous
Notes
Question
Suppose you save P at interest rate r , with interest compounded every
instant. How much do you have after 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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 24 / 38
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9.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
The magic number
Notes
Deﬁnition
n
1
e = lim 1+
n→∞ n
So now continuously-compounded interest can be expressed as
B(t) = Pe rt .
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 25 / 38
Existence of e
See Appendix B Notes
n
1
n 1+
We can experimentally verify n
that this number exists and 1 2
is 2 2.25
3 2.37037
e ≈ 2.718281828459045 . . . 10 2.59374
100 2.70481
e is irrational
1000 2.71692
e is transcendental 106 2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
Meet the Mathematician: Leonhard Euler
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 27 / 38
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10.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
A limit
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 28 / 38
Outline
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 29 / 38
Logarithms
Notes
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 e x . So
y = ln x ⇐⇒ x = e y .
Facts
(i) loga (x1 · x2 ) = loga x1 + loga x2
x1
(ii) loga = loga x1 − loga x2
x2
r
(iii) loga (x ) = r loga x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 38
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11.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Logarithms convert products to sums
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 31 / 38
Example
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 32 / 38
Graphs of logarithmic functions
Notes
y x xx x
y = yy = 3e = 2
10=y
y = log2 x
y = ln x
y = log3 x
(0, 1)
y = log10 x
(1, 0) x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 33 / 38
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12.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Change of base formula for exponentials
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 34 / 38
Example of changing base
Notes
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.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 35 / 38
Upshot of changing base
Notes
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
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 36 / 38
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13.
V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010
Summary
Notes
Exponentials turn sums into products
Logarithms turn products into sums
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 38 / 38
Notes
Notes
13
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