Lesson 13: Exponential and Logarithmic Functions (Section 021 handout)
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Transcript of Lesson 13: Exponential and Logarithmic Functions (Section 021 handout)
Sections 3.1–3.2Exponential and Logarithmic Functions
V63.0121.021, Calculus I
New York University
October 21, 2010
Announcements
I Midterm is graded and scores are on blackboard. Should get it backin recitation.
I There is WebAssign due Monday/Tuesday next week.
Announcements
I Midterm is graded andscores are on blackboard.Should get it back inrecitation.
I There is WebAssign dueMonday/Tuesday next week.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 2 / 38
Midterm Statistics
I Average: 78.77%
I Median: 80%
I Standard Deviation: 12.39%
I “good” is anything above average and “great” is anything more thanone standard deviation above average.
I 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
Notes
Notes
Notes
1
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
Objectives for Sections 3.1 and 3.2
I Know the definition of anexponential function
I Know the properties ofexponential functions
I Understand and apply thelaws of logarithms, includingthe change of base formula.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 4 / 38
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential functionCompound InterestThe number eA 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
Definition
If a is a real number and n is a positive whole number, then
an = a · a · · · · · a︸ ︷︷ ︸n factors
Examples
I 23 = 2 · 2 · 2 = 8
I 34 = 3 · 3 · 3 · 3 = 81
I (−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
Notes
Notes
Notes
2
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
Anatomy of a power
Definition
A power is an expression of the form ab.
I The number a is called the base.
I 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
If a is a real number, then
I ax+y = axay (sums to products)
I ax−y =ax
ay(differences to quotients)
I (ax)y = axy (repeated exponentiation to multiplied powers)
I (ab)x = axbx (power of product is product of powers)
whenever all exponents are positive whole numbers.
Proof.
Check for yourself:
ax+y = a · a · · · · · a︸ ︷︷ ︸x + y factors
= a · a · · · · · a︸ ︷︷ ︸x factors
· a · a · · · · · a︸ ︷︷ ︸y factors
= axay
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 8 / 38
Let’s be conventional
I The desire that these properties remain true gives us conventions forax when x is not a positive whole number.
I For example, what should a0 be? We cannot write down zero a’s andmultiply them together. But we would want this to be true:
an = an+0 != an · a0 =⇒ a0
!=
an
an= 1
(The equality with the exclamation point is what we want.)
Definition
If a 6= 0, we define a0 = 1.
I 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
Notes
Notes
Notes
3
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
Conventions for negative exponents
If n ≥ 0, we want
an+(−n) != an · a−n =⇒ a−n
!=
a0
an=
1
an
Definition
If n is a positive integer, we define a−n =1
an.
Fact
I The convention that a−n =1
an“works” for negative n as well.
I If m and n are any integers, then am−n =am
an.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 10 / 38
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 iseven.
Notice thatq√
ap =(
q√
a)p
. 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
I So ax is well-defined if a is positive and x is rational.
I What about irrational powers?
Definition
Let a > 0. Thenax = lim
r→xr rational
ar
In other words, to approximate ax for irrational x , take r close to x butrational and compute ar .
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 12 / 38
Notes
Notes
Notes
4
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
Approximating a power with an irrational exponent
r 2r
3 23 = 8
3.1 231/10 =10√
231 ≈ 8.57419
3.14 2314/100 =100√
2314 ≈ 8.81524
3.141 23141/1000 =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
x
y
y = 1x
y = 2xy = 3xy = 10x y = 1.5xy = (1/2)xy = (1/3)x y = (1/10)xy = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential functionCompound InterestThe number eA limit
Logarithmic Functions
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 15 / 38
Notes
Notes
Notes
5
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
Properties of exponential Functions
Theorem
If a > 0 and a 6= 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
I ax+y = axay
I ax−y =ax
ay(negative exponents mean reciprocals)
I (ax)y = axy (fractional exponents mean roots)I (ab)x = axbx
Proof.
I This is true for positive integer exponents by natural definitionI Our conventional definitions make these true for rational exponentsI 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
Example
Simplify: 82/3
Solution
I 82/3 =3√
82 =3√
64 = 4
I Or,(
3√
8)2
= 22 = 4.
Example
Simplify:
√8
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
Fact (Limits of exponentialfunctions)
I If a > 1, then limx→∞
ax =∞and lim
x→−∞ax = 0
I If 0 < a < 1, thenlimx→∞
ax = 0 and
limx→−∞
ax =∞ x
y
y = 1x
y = 2xy = 3xy = 10x y = 1.5xy = (1/2)xy = (1/3)x y = (1/10)xy = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 18 / 38
Notes
Notes
Notes
6
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential functionCompound InterestThe number eA limit
Logarithmic Functions
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 19 / 38
Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest compoundedonce a year. How much do you have
I After one year?
I After two years?
I after t years?
Answer
I $100 + 10% = $110
I $110 + 10% = $110 + $11 = $121
I $100(1.1)t .
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 20 / 38
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest compoundedfour times a year. How much do you have
I After one year?
I After two years?
I after t years?
Answer
I $100(1.025)4 = $110.38, not $100(1.1)4!
I $100(1.025)8 = $121.84
I $100(1.025)4t .
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
Notes
Notes
Notes
7
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
Compounded Interest: monthly
Question
Suppose you save $100 at 10% annual interest, with interest compoundedtwelve times a year. How much do you have after t years?
Answer
$100(1 + 10%/12)12t
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 22 / 38
Compounded Interest: general
Question
Suppose you save P at interest rate r , with interest compounded n times ayear. How much do you have after t years?
Answer
B(t) = P(
1 +r
n
)nt
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 23 / 38
Compounded Interest: continuous
Question
Suppose you save P at interest rate r , with interest compounded everyinstant. How much do you have after t years?
Answer
B(t) = limn→∞
P(
1 +r
n
)nt= lim
n→∞P
(1 +
1
n
)rnt
= P
[limn→∞
(1 +
1
n
)n
︸ ︷︷ ︸independent of P, r , or t
]rt
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 24 / 38
Notes
Notes
Notes
8
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
The magic number
Definition
e = limn→∞
(1 +
1
n
)n
So now continuously-compounded interest can be expressed as
B(t) = Pert .
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 25 / 38
Existence of eSee Appendix B
I We can experimentally verifythat this number exists andis
e ≈ 2.718281828459045 . . .
I e is irrational
I e is transcendental
n
(1 +
1
n
)n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
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
I Born in Switzerland, lived inPrussia (Germany) andRussia
I Eyesight trouble all his life,blind from 1766 onward
I Hundreds of contributions tocalculus, number theory,graph theory, fluidmechanics, optics, andastronomy
Leonhard Paul EulerSwiss, 1707–1783
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 27 / 38
Notes
Notes
Notes
9
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
A limit
Question
What is limh→0
eh − 1
h?
Answer
I e = limn→∞ (1 + 1/n)n = lim
h→0(1 + h)1/h. So for a small h, e ≈ (1 + h)1/h.
So
eh − 1
h≈[(1 + h)1/h
]h − 1
h= 1
I It follows that limh→0
eh − 1
h= 1.
I This can be used to characterize e: limh→0
2h − 1
h= 0.693 · · · < 1 and
limh→0
3h − 1
h= 1.099 · · · > 1
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 28 / 38
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential functionCompound InterestThe number eA limit
Logarithmic Functions
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 29 / 38
Logarithms
Definition
I The base a logarithm loga x is the inverse of the function ax
y = loga x ⇐⇒ x = ay
I The natural logarithm ln x is the inverse of ex . Soy = ln x ⇐⇒ x = ey .
Facts
(i) loga(x1 · x2) = loga x1 + loga x2
(ii) loga
(x1x2
)= loga x1 − loga x2
(iii) loga(x r ) = r loga x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 38
Notes
Notes
Notes
10
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
Logarithms convert products to sums
I Suppose y1 = loga x1 and y2 = loga x2I Then x1 = ay1 and x2 = ay2
I So x1x2 = ay1ay2 = ay1+y2
I Thereforeloga(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
Write as a single logarithm: 2 ln 4− ln 3.
Solution
I 2 ln 4− ln 3 = ln 42 − ln 3 = ln42
3
I notln 42
ln 3!
Example
Write as a single logarithm: ln3
4+ 4 ln 2
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
x
yy = 2x
y = log2 x
(0, 1)
(1, 0)
y = 3x
y = log3 x
y = 10x
y = log10 x
y = ex
y = ln x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 33 / 38
Notes
Notes
Notes
11
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
Change of base formula for exponentials
Fact
If a > 0 and a 6= 1, then
loga x =ln x
ln a
Proof.
I If y = loga x , then x = ay
I So ln x = ln(ay ) = y ln a
I Therefore
y = loga x =ln 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
Example
Find log2 8 by using log10 only.
Solution
log2 8 =log10 8
log10 2≈ 0.90309
0.30103= 3
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
The point of the change of base formula
loga x =logb x
logb a=
1
logb a· logb x = constant · logb x
is that all the logarithmic functions are multiples of each other. So justpick one and call it your favorite.
I Engineers like the common logarithm log = log10I Computer scientists like the binary logarithm lg = log2I Mathematicians like natural logarithm ln = loge
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 36 / 38
Notes
Notes
Notes
12
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010
Summary
I Exponentials turn sums into products
I Logarithms turn products into sums
I Slide rule scabbards are wicked cool
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 38 / 38
Notes
Notes
Notes
13
Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010