Lesson 13: Exponential and Logarithmic Functions (slides)

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
.

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

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.

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.

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

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

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

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.

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.

Fact
x−y ax
a = y (diﬀerences to quo ents)
a
(ax )y = axy
(ab)x = ax bx

Fact
x−y ax
a
(ax )y = axy (repeated exponen a on to mul plied powers)
(ab)x = ax bx

Fact
x−y ax
a
(ab)x = ax bx (power of product is product of powers)

Fact
x−y ax
a
(ab)x = ax bx (power of product is product of powers)

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
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

n
! ! a
an = an+0 = an · a0 =⇒ a0 = n = 1
a

n
! ! a
an = an+0 = an · a0 =⇒ a0 = n = 1
a

Deﬁni on
If a ̸= 0, we deﬁne a0 = 1.

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).

Conventions for negative exponents

If n ≥ 0, we want

an+(−n) = an · a−n
!


If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
= n
an a


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

Deﬁni on
1
an

Deﬁni on
1
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

Conventions for fractional exponents
If q is a posi ve integer, we want
!
(a1/q )q = a1 = a

! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a

! ! √
(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.

! ! √
(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

Conventions for irrational
exponents
So ax is well-deﬁned if a is posi ve and x is ra onal.
What about irra onal powers?

exponents

Deﬁni on
Let a > 0. Then
ax = lim ar
r→x
r ra onal

exponents

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 .

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

Graphs of exponential functions
y

. x

y

y = 1x
. x

y
y = 2x

y = 1x
. x

y
y = 3x = 2x
y

y = 1x
. x

y
y = 10x 3x = 2x
y= y

y = 1x
. x

y
y = 10x 3x = 2x
y= y y = 1.5x

y = 1x
. x

y
y = (1/2)x y = 10x 3x = 2x
y= y y = 1.5x

y = 1x
. x

y
y = (y/= x(1/3)x
1 2) y = 10x 3x = 2x
y= y y = 1.5x

y = 1x
. x

y
y = (y/= x(1/3)x
1 2) y = (1/10y x= 10x 3x = 2x
) y= y y = 1.5x

y = 1x
. x

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

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

Theorem
ax+y = ax ay
x−y ax
a = y (nega ve exponents mean reciprocals)
a
(a ) = axy
x y

(ab)x = ax bx

Theorem
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

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

Simplifying exponential expressions

Example
Simplify: 82/3


Example
Simplify: 82/3

Solu on
√
3
√
2/3
= 82 = 64 = 4
3
8


Example
Simplify: 82/3

Solu on
√3
√
2/3
8 = 82 = 64 = 4
3

(√ )2
8 = 22 = 4.
3
Or,

Simplifying exponential
expressions

Example
√
8
Simplify:
21/2

Simplifying exponential
expressions

Example
√
8
Simplify:
21/2

Answer
2

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→−∞

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?

Compounded Interest
Ques on

Answer
$100 + 10% = $110

Compounded Interest
Ques on

Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121

Compounded Interest
Ques on

Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
$100(1.1)t .

Compounded Interest: quarterly
Ques on
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?

Ques on

Answer
$100(1.025)4 = $110.38,

Ques on

Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !

Ques on

Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
$100(1.025)8 = $121.84

Ques on

Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
$100(1.025)8 = $121.84
$100(1.025)4t .

Compounded Interest: monthly

Ques on
compounded twelve mes a year. How much do you have a er t
years?

Compounded Interest: monthly

Ques on
compounded twelve mes a year. How much do you have a er t
years?

Answer
$100(1 + 10%/12)12t

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?

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

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?

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

The magic number
Deﬁni on
( )n
1
e = lim 1 +
n→∞ n

The magic number
Deﬁni on
( )n
1
e = lim 1 +
n→∞ n

So now con nuously-compounded interest can be expressed as

B(t) = Pert .

Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25

Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037

Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374

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

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

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

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

Existence of e
See Appendix B
( )n
1
n 1+
3 2.37037
e ≈ 2.718281828459045 . . .
10 2.59374
e is irra onal 100 2.70481
1000 2.71692
106 2.71828

Existence of e
See Appendix B
( )n
1
n 1+
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

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

A limit
Ques on
eh − 1
What is lim ?
h→0 h

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

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

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 .

Facts about Logarithms

Facts
(i) loga (x1 · x2 ) = loga x1 + loga x2


Facts
( )
x1
(ii) loga = loga x1 − loga x2
x2


Facts
( )
x1
(ii) loga = loga x1 − loga x2
x2
(iii) loga (xr ) = r loga x

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

Examples
Example
Write as a single logarithm: 2 ln 4 − ln 3.

Examples
Example
Write as a single logarithm: 2 ln 4 − ln 3.

Solu on
42
2 ln 4 − ln 3 = ln 42 − ln 3 = ln
3
ln 42
not !
ln 3

Examples
Example
3
Write as a single logarithm: ln + 4 ln 2
4

Examples
Example
3
Write as a single logarithm: ln + 4 ln 2
4

Solu on

3
ln + 4 ln 2 = ln 3 − ln 4 + 4 ln 2 = ln 3 − 2 ln 2 + 4 ln 2
4
= ln 3 + 2 ln 2 = ln(3 · 22 ) = ln 12

Graphs of logarithmic functions
y
y = 2x

y = log2 x

(0, 1)
.
(1, 0) x

y
y =y3= 2x
x

y = log2 x

y = log3 x
(0, 1)
.
(1, 0) x

y
y =y10y3= 2x
=x x

y = log2 x

y = log3 x
(0, 1)
y = log10 x
.
(1, 0) x

y
y =x ex
y =y10y3= 2x
= x

y = log2 x

yy= log3 x
= ln x
(0, 1)
y = log10 x
.
(1, 0) x

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

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

Example of changing base

Example
Find log2 8 by using log10 only.


Example

Solu on
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103


Example

Solu on
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised?


Example

Solu on
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised? No, log2 8 = log2 23 = 3 directly.

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.”

“lawn”
.

.

Image credit: Selva

Summary

Exponen als turn sums into products
Logarithms turn products into sums
Slide rule scabbards are wicked cool

Lesson 13: Exponential and Logarithmic Functions (slides)

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