Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
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Transcript of Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
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 itback in 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 nextweek.
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
than one 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
. . . . . .
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,including the change ofbase 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
DefinitionIf 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 = 8I 34 = 3 · 3 · 3 · 3 = 81I (−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
. . . . . .
Derivation of exponential functions
DefinitionIf 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 = 8I 34 = 3 · 3 · 3 · 3 = 81I (−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
. . . . . .
Anatomy of a power
DefinitionA 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
. . . . . .
FactIf 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
. . . . . .
FactIf 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
. . . . . .
FactIf 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
. . . . . .
FactIf 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
. . . . . .
FactIf 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
. . . . . .
FactIf 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 conventionsfor ax when x is not a positive whole number.
I For example, what should a0 be? We cannot write down zero a’sand multiply 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.)
DefinitionIf a ̸= 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
. . . . . .
Let's be conventional
I The desire that these properties remain true gives us conventionsfor ax when x is not a positive whole number.
I For example, what should a0 be? We cannot write down zero a’sand multiply 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.)
DefinitionIf a ̸= 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
. . . . . .
Let's be conventional
I The desire that these properties remain true gives us conventionsfor ax when x is not a positive whole number.
I For example, what should a0 be? We cannot write down zero a’sand multiply 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.)
DefinitionIf a ̸= 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
. . . . . .
Let's be conventional
I The desire that these properties remain true gives us conventionsfor ax when x is not a positive whole number.
I For example, what should a0 be? We cannot write down zero a’sand multiply 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.)
DefinitionIf a ̸= 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
. . . . . .
Let's be conventional
I The desire that these properties remain true gives us conventionsfor ax when x is not a positive whole number.
I For example, what should a0 be? We cannot write down zero a’sand multiply 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.)
DefinitionIf a ̸= 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
. . . . . .
Conventions for negative exponents
If n ≥ 0, we want
an+(−n) != an · a−n
=⇒ a−n !=
a0
an =1an
Definition
If n is a positive integer, we define a−n =1an .
Fact
I The convention that a−n =1an “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 negative exponents
If n ≥ 0, we want
an+(−n) != an · a−n =⇒ a−n !
=a0
an =1an
Definition
If n is a positive integer, we define a−n =1an .
Fact
I The convention that a−n =1an “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 negative exponents
If n ≥ 0, we want
an+(−n) != an · a−n =⇒ a−n !
=a0
an =1an
Definition
If n is a positive integer, we define a−n =1an .
Fact
I The convention that a−n =1an “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 negative exponents
If n ≥ 0, we want
an+(−n) != an · a−n =⇒ a−n !
=a0
an =1an
Definition
If n is a positive integer, we define a−n =1an .
Fact
I The convention that a−n =1an “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
DefinitionIf q is a positive integer, we define a1/q = q
√a. We must have a ≥ 0 if q
is even.
Notice that q√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 fractional exponents
If q is a positive integer, we want
(a1/q)q != a1 = a =⇒ a1/q !
= q√a
DefinitionIf q is a positive integer, we define a1/q = q
√a. We must have a ≥ 0 if q
is even.
Notice that q√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 fractional exponents
If q is a positive integer, we want
(a1/q)q != a1 = a =⇒ a1/q !
= q√a
DefinitionIf q is a positive integer, we define a1/q = q
√a. We must have a ≥ 0 if q
is even.
Notice that q√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 fractional exponents
If q is a positive integer, we want
(a1/q)q != a1 = a =⇒ a1/q !
= q√a
DefinitionIf q is a positive integer, we define a1/q = q
√a. We must have a ≥ 0 if q
is even.
Notice that q√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?
DefinitionLet a > 0. Then
ax = limr→x
r 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
. . . . . .
Conventions for irrational exponents
I So ax is well-defined if a is positive and x is rational.I What about irrational powers?
DefinitionLet a > 0. Then
ax = limr→x
r 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
. . . . . .
Conventions for irrational exponents
I So ax is well-defined if a is positive and x is rational.I What about irrational powers?
DefinitionLet a > 0. Then
ax = limr→x
r 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
. . . . . .
Approximating a power with an irrational exponent
r 2r
3 23 = 83.1 231/10 =
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 = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
. . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
. . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x
.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
. . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x.y = 3x
.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
. . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x.y = 3x.y = 10x
.y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
. . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x.y = 3x.y = 10x .y = 1.5x
.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
. . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x
.y = (1/3)x .y = (1/10)x.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
. . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x
.y = (1/10)x.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
. . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x
.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
. . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (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
. . . . . .
Properties of exponential Functions.
.
TheoremIf a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain(−∞,∞) and range (0,∞). In particular, ax > 0 for all x. For any real numbersx 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
. . . . . .
Properties of exponential Functions.
.
TheoremIf a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain(−∞,∞) and range (0,∞). In particular, ax > 0 for all x. For any real numbersx 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
. . . . . .
Properties of exponential Functions.
.
TheoremIf a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain(−∞,∞) and range (0,∞). In particular, ax > 0 for all x. For any real numbersx 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
Answer2
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 17 / 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
Answer2
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 17 / 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
Answer2
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 17 / 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
Answer2
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 17 / 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
Answer2V63.0121.021, 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 andlim
x→−∞ax = ∞ . .x
.y
.y = 1x
.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 18 / 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 19 / 38
. . . . . .
Compounded Interest
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded once a year. How much do you have
I After one year?I After two years?I after t years?
Answer
I $100+ 10% = $110I $110+ 10% = $110+ $11 = $121I $100(1.1)t.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 20 / 38
. . . . . .
Compounded Interest
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded once 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 = $121I $100(1.1)t.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 20 / 38
. . . . . .
Compounded Interest
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded once a year. How much do you have
I After one year?I After two years?I after t years?
Answer
I $100+ 10% = $110I $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
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded once a year. How much do you have
I After one year?I After two years?I after t years?
Answer
I $100+ 10% = $110I $110+ 10% = $110+ $11 = $121I $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
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded four 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 = $121.84I $100(1.025)4t.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
. . . . . .
Compounded Interest: quarterly
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded four 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 = $121.84I $100(1.025)4t.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
. . . . . .
Compounded Interest: quarterly
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded four 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 = $121.84I $100(1.025)4t.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
. . . . . .
Compounded Interest: quarterly
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded four 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
. . . . . .
Compounded Interest: quarterly
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded four 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 = $121.84I $100(1.025)4t.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
. . . . . .
Compounded Interest: monthly
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded twelve times a year. How much do you have after tyears?
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: monthly
QuestionSuppose you save $100 at 10% annual interest, with interestcompounded twelve times a year. How much do you have after tyears?
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
QuestionSuppose you save P at interest rate r, with interest compounded ntimes a year. How much do you have after t years?
Answer
B(t) = P(1+
rn
)nt
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 23 / 38
. . . . . .
Compounded Interest: general
QuestionSuppose you save P at interest rate r, with interest compounded ntimes a year. How much do you have after t years?
Answer
B(t) = P(1+
rn
)nt
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 23 / 38
. . . . . .
Compounded Interest: continuous
QuestionSuppose 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+
rn
)nt= lim
n→∞P(1+
1n
)rnt
= P[
limn→∞
(1+
1n
)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
. . . . . .
Compounded Interest: continuous
QuestionSuppose 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+
rn
)nt= lim
n→∞P(1+
1n
)rnt
= P[
limn→∞
(1+
1n
)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
. . . . . .
The magic number
Definition
e = limn→∞
(1+
1n
)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
. . . . . .
The magic number
Definition
e = limn→∞
(1+
1n
)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 experimentallyverify that this numberexists and is
e ≈ 2.718281828459045 . . .
I e is irrationalI e is transcendental
n(1+
1n
)n
1 22 2.25
3 2.3703710 2.59374100 2.704811000 2.71692106 2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
. . . . . .
Existence of eSee Appendix B
I We can experimentallyverify that this numberexists and is
e ≈ 2.718281828459045 . . .
I e is irrationalI e is transcendental
n(1+
1n
)n
1 22 2.253 2.37037
10 2.59374100 2.704811000 2.71692106 2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
. . . . . .
Existence of eSee Appendix B
I We can experimentallyverify that this numberexists and is
e ≈ 2.718281828459045 . . .
I e is irrationalI e is transcendental
n(1+
1n
)n
1 22 2.253 2.3703710 2.59374
100 2.704811000 2.71692106 2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
. . . . . .
Existence of eSee Appendix B
I We can experimentallyverify that this numberexists and is
e ≈ 2.718281828459045 . . .
I e is irrationalI e is transcendental
n(1+
1n
)n
1 22 2.253 2.3703710 2.59374100 2.70481
1000 2.71692106 2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
. . . . . .
Existence of eSee Appendix B
I We can experimentallyverify that this numberexists and is
e ≈ 2.718281828459045 . . .
I e is irrationalI e is transcendental
n(1+
1n
)n
1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692
106 2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
. . . . . .
Existence of eSee Appendix B
I We can experimentallyverify that this numberexists and is
e ≈ 2.718281828459045 . . .
I e is irrationalI e is transcendental
n(1+
1n
)n
1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
. . . . . .
Existence of eSee Appendix B
I We can experimentallyverify that this numberexists and is
e ≈ 2.718281828459045 . . .
I e is irrationalI e is transcendental
n(1+
1n
)n
1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
. . . . . .
Existence of eSee Appendix B
I We can experimentallyverify that this numberexists and is
e ≈ 2.718281828459045 . . .
I e is irrational
I e is transcendental
n(1+
1n
)n
1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
. . . . . .
Existence of eSee Appendix B
I We can experimentallyverify that this numberexists and is
e ≈ 2.718281828459045 . . .
I e is irrationalI e is transcendental
n(1+
1n
)n
1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 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, livedin Prussia (Germany) andRussia
I Eyesight trouble all his life,blind from 1766 onward
I Hundreds of contributionsto calculus, 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
. . . . . .
A limit.
.
Question
What is limh→0
eh − 1h
?
Answer
I e = limn→∞
(1+ 1/n)n = limh→0
(1+ h)1/h. So for a small h, e ≈ (1+ h)1/h. So
eh − 1h
≈[(1+ h)1/h
]h − 1h
= 1
I It follows that limh→0
eh − 1h
= 1.
I This can be used to characterize e: limh→0
2h − 1h
= 0.693 · · · < 1 and
limh→0
3h − 1h
= 1.099 · · · > 1
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 28 / 38
. . . . . .
A limit.
.
Question
What is limh→0
eh − 1h
?
Answer
I e = limn→∞
(1+ 1/n)n = limh→0
(1+ h)1/h. So for a small h, e ≈ (1+ h)1/h. So
eh − 1h
≈[(1+ h)1/h
]h − 1h
= 1
I It follows that limh→0
eh − 1h
= 1.
I This can be used to characterize e: limh→0
2h − 1h
= 0.693 · · · < 1 and
limh→0
3h − 1h
= 1.099 · · · > 1
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 28 / 38
. . . . . .
A limit.
.
Question
What is limh→0
eh − 1h
?
Answer
I e = limn→∞
(1+ 1/n)n = limh→0
(1+ h)1/h. So for a small h, e ≈ (1+ h)1/h. So
eh − 1h
≈[(1+ h)1/h
]h − 1h
= 1
I It follows that limh→0
eh − 1h
= 1.
I This can be used to characterize e: limh→0
2h − 1h
= 0.693 · · · < 1 and
limh→0
3h − 1h
= 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(xr) = r loga x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 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(xr) = r loga x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 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(xr) = r loga x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 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(xr) = r loga x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 38
. . . . . .
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: ln34+ 4 ln 2
Answerln 12
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 32 / 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: ln34+ 4 ln 2
Answerln 12
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 32 / 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: ln34+ 4 ln 2
Answerln 12
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 32 / 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: ln34+ 4 ln 2
Answerln 12
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 32 / 38
. . . . . .
Graphs of logarithmic functions
. .x
.y.y = 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
. . . . . .
Graphs of logarithmic functions
. .x
.y.y = 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
. . . . . .
Graphs of logarithmic functions
. .x
.y.y = 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
. . . . . .
Graphs of logarithmic functions
. .x
.y.y = 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
. . . . . .
Change of base formula for exponentials
FactIf a > 0 and a ̸= 1, and the same for b, then
loga x =logb xlogb a
Proof.
I If y = loga x, then x = ay
I So logb x = logb(ay) = y logb a
I Thereforey = loga x =
logb xlogb a
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 34 / 38
. . . . . .
Change of base formula for exponentials
FactIf a > 0 and a ̸= 1, and the same for b, then
loga x =logb xlogb a
Proof.
I If y = loga x, then x = ay
I So logb x = logb(ay) = y logb a
I Thereforey = loga x =
logb xlogb 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 8log10 2
≈ 0.903090.30103
= 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
. . . . . .
Example of changing base
Example
Find log2 8 by using log10 only.
Solution
log2 8 =log10 8log10 2
≈ 0.903090.30103
= 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
. . . . . .
Example of changing base
Example
Find log2 8 by using log10 only.
Solution
log2 8 =log10 8log10 2
≈ 0.903090.30103
= 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
. . . . . .
Example of changing base
Example
Find log2 8 by using log10 only.
Solution
log2 8 =log10 8log10 2
≈ 0.903090.30103
= 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 xlogb 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
Naturally, we will follow the mathematicians. Just don’t pronounce it“lawn.”
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 36 / 38
. . . . . .
..“lawn”
.
.Image credit: SelvaV63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 37 / 38