Lesson 13: Exponential and Logarithmic Functions (Section 021 handout)

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.


1


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.


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


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


2


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.


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


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).


3


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


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


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 .


4


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


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


Outline
Notes



Compound Interest
The number e
A limit



5


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


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

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→−∞


6


Outline
Notes



Compound Interest
The number e
A limit



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


Compounded Interest: quarterly
Notes

Question
four times a year. How much do you have
After one year?
After two years?
after t years?

Answer


7


Compounded Interest: monthly
Notes

Question
twelve times a year. How much do you have after t years?

Answer


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


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


8


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 .


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


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

9


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


Outline
Notes



Compound Interest
The number e
A limit



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


10


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


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


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


11


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


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.


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


12


Summary
Notes

Exponentials turn sums into products
Logarithms turn products into sums


Notes

Notes

13

Lesson 13: Exponential and Logarithmic Functions (Section 021 handout)

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