Definitions and elementary properties of exponential and logarithmic functions.

1. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Notes
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
.
.
Notes
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
.
.
Notes
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.
.
.
. 1
.

2. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Notes
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
.
.
Notes
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
.
.
Notes
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.
.
.
. 2
.

3. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Fact Notes
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quo ents)
a
(ax )y = axy (repeated exponen a on to mul plied powers)
(ab)x = ax bx (power of product is product of powers)
whenever all exponents are posi ve 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
.
Notes
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:
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).
.
.
Notes
Conventions for negative exponents
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
.
.
. 3
.

4. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Notes
Defini on
1
If n is a posi ve integer, we define a−n = .
an
Fact
1
The conven on that a−n = “works” for nega ve n as well.
an
am
If m and n are any integers, then am−n = n .
a
.
.
Notes
Conventions for fractional exponents
If q is a posi ve integer, we want
! ! √
(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 Notes
exponents
So ax is well-defined if a is posi ve and x is ra onal.
What about irra onal powers?
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 .
.
.
. 4
.

5. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Approximating a power with an Notes
irrational exponent
r 2r
3
3 √=8
2
10
3.1 231/10 = √ 31 ≈ 8.57419
2
100
3.14 2 314/100
= √ 314 ≈ 8.81524
2
1000
3.141 23141/1000 = 23141 ≈ 8.82135
The limit (numerically approximated is)
2π ≈ 8.82498
.
.
Graphs of various exponential Notes
functions x x x y x x x
y y =y/=3(1/3)
= (1(2/ )
2)
x y = (1/10y = 10 3 = 2
) y= y y = 1.5x
y = 1x
. x
.
.
Notes
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
.
.
. 5
.

6. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Notes
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
ax
ax−y = y (nega ve exponents mean reciprocals)
a
(ax )y = axy (frac onal exponents mean roots)
(ab)x = ax bx
.
.
Notes
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 Notes
expressions
Example
Simplify: 82/3
Solu on
√3
√
82/3 = 82 = 64 = 4
3
(√ )2
8 = 22 = 4.
3
Or,
.
.
. 6
.

7. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Simplifying exponential Notes
expressions
Example
√
8
Simplify:
21/2
Answer
2
.
.
Notes
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 = 1.5x
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→−∞
.
.
Notes
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
.
.
. 7
.

8. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Notes
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?
Answer
.
.
Notes
Compounded Interest: quarterly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded four mes a year. How much do you have A er one
year? A er two years? A er t years?
Answer
.
.
Notes
Compounded Interest: monthly
Ques on
Suppose you save $100 at 10% annual interest, with interest
compounded twelve mes a year. How much do you have a er t
years?
Answer
.
.
. 8
.

9. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Notes
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
.
.
Notes
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
.
.
Notes
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 .
.
.
. 9
.

10. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Existence of e Notes
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
e is irra onal 100 2.70481
1000 2.71692
e is transcendental
106 2.71828
.
.
Notes
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
.
.
Notes
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
.
.
. 10
.

11. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Notes
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
.
.
Notes
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
.
.
Notes
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 .
.
.
. 11
.

12. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Notes
Facts about Logarithms
Facts
(i) loga (x1 · x2 ) = loga x1 + loga x2
( )
x1
(ii) loga = loga x1 − loga x2
x2
r
(iii) loga (x ) = r loga x
.
.
Notes
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
.
.
Notes
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
.
.
. 12
.

13. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Notes
Examples
Example
3
Write as a single logarithm: ln + 4 ln 2
4
Answer
ln 12
.
.
Notes
Graphs of logarithmic functions
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 Notes
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
.
.
. 13
.

14. . V63.0121.001: Calculus I
. Sec on 3.1–3.2: Exponen al Func ons
. March 9, 2011
Notes
Example of changing base
Example
Find log2 8 by using log10 only.
Solu on
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised? No, log2 8 = log2 23 = 3 directly.
.
.
Notes
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.”
.
.
Notes
Summary
Exponen als turn sums into products
Logarithms turn products into sums
.
.
. 14
.