lesson 13: exponential and logarithmic functions (slides)

107
. . SecƟon 3.1–3.2 ExponenƟal and Logarithmic FuncƟons V63.0121.001: Calculus I Professor MaƩhew Leingang New York University March 9, 2011

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Definitions and elementary properties of exponential and logarithmic functions.

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Page 1: Lesson 13: Exponential and Logarithmic Functions (slides)

..

Sec on 3.1–3.2Exponen al and Logarithmic

Func ons

V63.0121.001: Calculus IProfessor Ma hew Leingang

New York University

March 9, 2011

Page 2: Lesson 13: Exponential and Logarithmic Functions (slides)

Announcements

I Midterm is graded.average = 44, median=46,SD =10

I There is WebAssign duea er Spring Break.

I Quiz 3 on 2.6, 2.8, 3.1, 3.2on March 30

Page 3: Lesson 13: Exponential and Logarithmic Functions (slides)

Midterm Statistics

I Average: 43.86/60 = 73.1%I Median: 46/60 = 76.67%I Standard Devia on: 10.64%I “good” is anything above average and “great” is anything morethan one standard devia on above average.

I More than one SD below the mean is cause for concern.

Page 4: Lesson 13: Exponential and Logarithmic Functions (slides)

Objectives for Sections 3.1 and 3.2

I Know the defini on of anexponen al func on

I Know the proper es ofexponen al func ons

I Understand and applythe laws of logarithms,including the change ofbase formula.

Page 5: Lesson 13: Exponential and Logarithmic Functions (slides)

OutlineDefini on of exponen al func ons

Proper es of exponen al Func ons

The number e and the natural exponen al func onCompound InterestThe number eA limit

Logarithmic Func ons

Page 6: Lesson 13: Exponential and Logarithmic Functions (slides)

Derivation of exponentialsDefini onIf a is a real number and n is a posi ve 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

Page 7: Lesson 13: Exponential and Logarithmic Functions (slides)

Derivation of exponentialsDefini onIf a is a real number and n is a posi ve 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

Page 8: Lesson 13: Exponential and Logarithmic Functions (slides)

Anatomy of a power

Defini onA power is an expression of the form ab.

I The number a is called the base.I The number b is called the exponent.

Page 9: Lesson 13: Exponential and Logarithmic Functions (slides)

FactIf a is a real number, then

I ax+y = axay (sums to products)

I ax−y =ax

ay

(differences to quo ents)

I (ax)y = axy

(repeated exponen a on to mul plied powers)

I (ab)x = axbx

(power of product is product of powers)

whenever all exponents are posi ve 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

Page 10: Lesson 13: Exponential and Logarithmic Functions (slides)

FactIf a is a real number, then

I ax+y = axay (sums to products)

I ax−y =ax

ay (differences to quo ents)

I (ax)y = axy

(repeated exponen a on to mul plied powers)

I (ab)x = axbx

(power of product is product of powers)

whenever all exponents are posi ve 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

Page 11: Lesson 13: Exponential and Logarithmic Functions (slides)

FactIf a is a real number, then

I ax+y = axay (sums to products)

I ax−y =ax

ay (differences to quo ents)

I (ax)y = axy (repeated exponen a on to mul plied powers)I (ab)x = axbx

(power of product is product of powers)

whenever all exponents are posi ve 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

Page 12: Lesson 13: Exponential and Logarithmic Functions (slides)

FactIf a is a real number, then

I ax+y = axay (sums to products)

I ax−y =ax

ay (differences to quo ents)

I (ax)y = axy (repeated exponen a on to mul plied powers)I (ab)x = axbx (power of product is product of powers)

whenever all exponents are posi ve 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

Page 13: Lesson 13: Exponential and Logarithmic Functions (slides)

FactIf a is a real number, then

I ax+y = axay (sums to products)

I ax−y =ax

ay (differences to quo ents)

I (ax)y = axy (repeated exponen a on to mul plied powers)I (ab)x = axbx (power of product is product of powers)

whenever all exponents are posi ve 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

Page 14: Lesson 13: Exponential and Logarithmic Functions (slides)

FactIf a is a real number, then

I ax+y = axay (sums to products)

I ax−y =ax

ay (differences to quo ents)

I (ax)y = axy (repeated exponen a on to mul plied powers)I (ab)x = axbx (power of product is product of powers)

whenever all exponents are posi ve 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

Page 15: Lesson 13: Exponential and Logarithmic Functions (slides)

Let’s be conventionalI The desire that these proper es remain true gives usconven ons for ax when x is not a posi ve whole number.

I For example, what should a0 be?We would want this to be true:

an = an+0 != an · a0 =⇒ a0 !

=an

an = 1

Defini onIf a ̸= 0, we define a0 = 1.

I No ce 00 remains undefined (as a limit form, it’sindeterminate).

Page 16: Lesson 13: Exponential and Logarithmic Functions (slides)

Let’s be conventionalI The desire that these proper es remain true gives usconven ons for ax when x is not a posi ve whole number.

I For example, what should a0 be?We would want this to be true:

an = an+0 != an · a0

=⇒ a0 !=

an

an = 1

Defini onIf a ̸= 0, we define a0 = 1.

I No ce 00 remains undefined (as a limit form, it’sindeterminate).

Page 17: Lesson 13: Exponential and Logarithmic Functions (slides)

Let’s be conventionalI The desire that these proper es remain true gives usconven ons for ax when x is not a posi ve whole number.

I For example, what should a0 be?We would want this to be true:

an = an+0 != an · a0 =⇒ a0 !

=an

an = 1

Defini onIf a ̸= 0, we define a0 = 1.

I No ce 00 remains undefined (as a limit form, it’sindeterminate).

Page 18: Lesson 13: Exponential and Logarithmic Functions (slides)

Let’s be conventionalI The desire that these proper es remain true gives usconven ons for ax when x is not a posi ve whole number.

I For example, what should a0 be?We would want this to be true:

an = an+0 != an · a0 =⇒ a0 !

=an

an = 1

Defini onIf a ̸= 0, we define a0 = 1.

I No ce 00 remains undefined (as a limit form, it’sindeterminate).

Page 19: Lesson 13: Exponential and Logarithmic Functions (slides)

Let’s be conventionalI The desire that these proper es remain true gives usconven ons for ax when x is not a posi ve whole number.

I For example, what should a0 be?We would want this to be true:

an = an+0 != an · a0 =⇒ a0 !

=an

an = 1

Defini onIf a ̸= 0, we define a0 = 1.

I No ce 00 remains undefined (as a limit form, it’sindeterminate).

Page 20: Lesson 13: Exponential and Logarithmic Functions (slides)

Conventions for negative exponents

If n ≥ 0, we want

an+(−n) != an · a−n

=⇒ a−n !=

a0

an =1an

Defini on

If n is a posi ve integer, we define a−n =1an .

Page 21: Lesson 13: Exponential and Logarithmic Functions (slides)

Conventions for negative exponents

If n ≥ 0, we want

an+(−n) != an · a−n =⇒ a−n !

=a0

an =1an

Defini on

If n is a posi ve integer, we define a−n =1an .

Page 22: Lesson 13: Exponential and Logarithmic Functions (slides)

Conventions for negative exponents

If n ≥ 0, we want

an+(−n) != an · a−n =⇒ a−n !

=a0

an =1an

Defini on

If n is a posi ve integer, we define a−n =1an .

Page 23: Lesson 13: Exponential and Logarithmic Functions (slides)

Defini on

If n is a posi ve integer, we define a−n =1an .

Fact

I The conven on that a−n =1an “works” for nega ve n as well.

I If m and n are any integers, then am−n =am

an .

Page 24: Lesson 13: Exponential and Logarithmic Functions (slides)

Defini on

If n is a posi ve integer, we define a−n =1an .

Fact

I The conven on that a−n =1an “works” for nega ve n as well.

I If m and n are any integers, then am−n =am

an .

Page 25: Lesson 13: Exponential and Logarithmic Functions (slides)

Conventions for fractional exponentsIf q is a posi ve integer, we want

(a1/q)q != a1 = a

=⇒ a1/q != q

√a

Defini onIf q is a posi ve integer, we define a1/q = q

√a. We must have a ≥ 0

if q is even.

No ce that q√ap =

(q√a)p. So we can unambiguously say

ap/q = (ap)1/q = (a1/q)p

Page 26: Lesson 13: Exponential and Logarithmic Functions (slides)

Conventions for fractional exponentsIf q is a posi ve integer, we want

(a1/q)q != a1 = a =⇒ a1/q !

= q√a

Defini onIf q is a posi ve integer, we define a1/q = q

√a. We must have a ≥ 0

if q is even.

No ce that q√ap =

(q√a)p. So we can unambiguously say

ap/q = (ap)1/q = (a1/q)p

Page 27: Lesson 13: Exponential and Logarithmic Functions (slides)

Conventions for fractional exponentsIf q is a posi ve integer, we want

(a1/q)q != a1 = a =⇒ a1/q !

= q√a

Defini onIf q is a posi ve integer, we define a1/q = q

√a. We must have a ≥ 0

if q is even.

No ce that q√ap =

(q√a)p. So we can unambiguously say

ap/q = (ap)1/q = (a1/q)p

Page 28: Lesson 13: Exponential and Logarithmic Functions (slides)

Conventions for fractional exponentsIf q is a posi ve integer, we want

(a1/q)q != a1 = a =⇒ a1/q !

= q√a

Defini onIf q is a posi ve integer, we define a1/q = q

√a. We must have a ≥ 0

if q is even.

No ce that q√ap =

(q√a)p. So we can unambiguously say

ap/q = (ap)1/q = (a1/q)p

Page 29: Lesson 13: Exponential and Logarithmic Functions (slides)

Conventions for irrationalexponents

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

Defini onLet a > 0. Then

ax = limr→x

r ra onalar

In other words, to approximate ax for irra onal x, take r close to xbut ra onal and compute ar.

Page 30: Lesson 13: Exponential and Logarithmic Functions (slides)

Conventions for irrationalexponents

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

Defini onLet a > 0. Then

ax = limr→x

r ra onalar

In other words, to approximate ax for irra onal x, take r close to xbut ra onal and compute ar.

Page 31: Lesson 13: Exponential and Logarithmic Functions (slides)

Conventions for irrationalexponents

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

Defini onLet a > 0. Then

ax = limr→x

r ra onalar

In other words, to approximate ax for irra onal x, take r close to xbut ra onal and compute ar.

Page 32: Lesson 13: Exponential and Logarithmic Functions (slides)

Approximating a power with anirrational exponent

r 2r

3 23 = 83.1 231/10 = 10

√231 ≈ 8.57419

3.14 2314/100 = 100√

2314 ≈ 8.815243.141 23141/1000 = 1000

√23141 ≈ 8.82135

The limit (numerically approximated is)

2π ≈ 8.82498

Page 33: Lesson 13: Exponential and Logarithmic Functions (slides)

Graphs of exponential functions

.. x.

y

Page 34: Lesson 13: Exponential and Logarithmic Functions (slides)

Graphs of exponential functions

.. x.

y

.y = 1x

Page 35: Lesson 13: Exponential and Logarithmic Functions (slides)

Graphs of exponential functions

.. x.

y

.y = 1x

.

y = 2x

Page 36: Lesson 13: Exponential and Logarithmic Functions (slides)

Graphs of exponential functions

.. x.

y

.y = 1x

.

y = 2x

.

y = 3x

Page 37: Lesson 13: Exponential and Logarithmic Functions (slides)

Graphs of exponential functions

.. x.

y

.y = 1x

.

y = 2x

.

y = 3x

.

y = 10x

Page 38: Lesson 13: Exponential and Logarithmic Functions (slides)

Graphs of exponential functions

.. x.

y

.y = 1x

.

y = 2x

.

y = 3x

.

y = 10x

.

y = 1.5x

Page 39: Lesson 13: Exponential and Logarithmic Functions (slides)

Graphs of exponential functions

.. x.

y

.y = 1x

.

y = 2x

.

y = 3x

.

y = 10x

.

y = 1.5x

.

y = (1/2)x

Page 40: Lesson 13: Exponential and Logarithmic Functions (slides)

Graphs of exponential functions

.. x.

y

.y = 1x

.

y = 2x

.

y = 3x

.

y = 10x

.

y = 1.5x

.

y = (1/2)x

.

y = (1/3)x

Page 41: Lesson 13: Exponential and Logarithmic Functions (slides)

Graphs of 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

Page 42: Lesson 13: Exponential and Logarithmic Functions (slides)

Graphs of 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

Page 43: Lesson 13: Exponential and Logarithmic Functions (slides)

OutlineDefini on of exponen al func ons

Proper es of exponen al Func ons

The number e and the natural exponen al func onCompound InterestThe number eA limit

Logarithmic Func ons

Page 44: Lesson 13: Exponential and Logarithmic Functions (slides)

Properties of exponential Functions

TheoremIf a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on withdomain (−∞,∞) 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

I ax+y = axay

I ax−y =ax

ay

(nega ve exponents mean reciprocals)

I (ax)y = axy

(frac onal exponents mean roots)

I (ab)x = axbx

Page 45: Lesson 13: Exponential and Logarithmic Functions (slides)

Properties of exponential Functions

TheoremIf a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on withdomain (−∞,∞) 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

I ax+y = axay

I ax−y =ax

ay (nega ve exponents mean reciprocals)

I (ax)y = axy

(frac onal exponents mean roots)

I (ab)x = axbx

Page 46: Lesson 13: Exponential and Logarithmic Functions (slides)

Properties of exponential Functions

TheoremIf a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on withdomain (−∞,∞) 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

I ax+y = axay

I ax−y =ax

ay (nega ve exponents mean reciprocals)

I (ax)y = axy (frac onal exponents mean roots)I (ab)x = axbx

Page 47: Lesson 13: Exponential and Logarithmic Functions (slides)

Proof.

I This is true for posi ve integer exponents by natural defini onI Our conven onal defini ons make these true for ra onalexponents

I Our limit defini on make these for irra onal exponents, too

Page 48: Lesson 13: Exponential and Logarithmic Functions (slides)

Simplifying exponential expressions

Example

Simplify: 82/3

Solu on

I 82/3 = 3√

82 = 3√64 = 4

I Or,(

3√8)2

= 22 = 4.

Page 49: Lesson 13: Exponential and Logarithmic Functions (slides)

Simplifying exponential expressions

Example

Simplify: 82/3

Solu on

I 82/3 = 3√

82 = 3√64 = 4

I Or,(

3√8)2

= 22 = 4.

Page 50: Lesson 13: Exponential and Logarithmic Functions (slides)

Simplifying exponential expressions

Example

Simplify: 82/3

Solu on

I 82/3 = 3√

82 = 3√64 = 4

I Or,(

3√8)2

= 22 = 4.

Page 51: Lesson 13: Exponential and Logarithmic Functions (slides)

Simplifying exponentialexpressions

Example

Simplify:√8

21/2

Answer2

Page 52: Lesson 13: Exponential and Logarithmic Functions (slides)

Simplifying exponentialexpressions

Example

Simplify:√8

21/2

Answer2

Page 53: Lesson 13: Exponential and Logarithmic Functions (slides)

Limits of exponential functionsFact (Limits of exponen alfunc ons)

I If a > 1, thenlimx→∞

ax = ∞ andlim

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

Page 54: Lesson 13: Exponential and Logarithmic Functions (slides)

OutlineDefini on of exponen al func ons

Proper es of exponen al Func ons

The number e and the natural exponen al func onCompound InterestThe number eA limit

Logarithmic Func ons

Page 55: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded InterestQues on

Suppose you save $100 at 10% annual interest, with interestcompounded once a year. How much do you have A er one year?A er two years? A er t years?

Answer

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

Page 56: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded InterestQues on

Suppose you save $100 at 10% annual interest, with interestcompounded once a year. How much do you have A er one year?A er two years? A er t years?

Answer

I $100+ 10% = $110

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

Page 57: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded InterestQues on

Suppose you save $100 at 10% annual interest, with interestcompounded once a year. How much do you have A er one year?A er two years? A er t years?

Answer

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

I $100(1.1)t.

Page 58: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded InterestQues on

Suppose you save $100 at 10% annual interest, with interestcompounded once a year. How much do you have A er one year?A er two years? A er t years?

Answer

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

Page 59: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: quarterlyQues on

Suppose you save $100 at 10% annual interest, with interestcompounded four mes a year. How much do you have A er oneyear? A er two years? A er 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.

Page 60: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: quarterlyQues on

Suppose you save $100 at 10% annual interest, with interestcompounded four mes a year. How much do you have A er oneyear? A er two years? A er 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.

Page 61: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: quarterlyQues on

Suppose you save $100 at 10% annual interest, with interestcompounded four mes a year. How much do you have A er oneyear? A er two years? A er 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.

Page 62: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: quarterlyQues on

Suppose you save $100 at 10% annual interest, with interestcompounded four mes a year. How much do you have A er oneyear? A er two years? A er 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.

Page 63: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: quarterlyQues on

Suppose you save $100 at 10% annual interest, with interestcompounded four mes a year. How much do you have A er oneyear? A er two years? A er 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.

Page 64: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: monthly

Ques on

Suppose you save $100 at 10% annual interest, with interestcompounded twelve mes a year. How much do you have a er tyears?

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

Page 65: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: monthly

Ques on

Suppose you save $100 at 10% annual interest, with interestcompounded twelve mes a year. How much do you have a er tyears?

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

Page 66: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: general

Ques on

Suppose you save P at interest rate r, with interest compounded nmes a year. How much do you have a er t years?

Answer

B(t) = P(1+

rn

)nt

Page 67: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: general

Ques on

Suppose you save P at interest rate r, with interest compounded nmes a year. How much do you have a er t years?

Answer

B(t) = P(1+

rn

)nt

Page 68: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: continuousQues on

Suppose you save P at interest rate r, with interest compoundedevery instant. How much do you have a er 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

Page 69: Lesson 13: Exponential and Logarithmic Functions (slides)

Compounded Interest: continuousQues on

Suppose you save P at interest rate r, with interest compoundedevery instant. How much do you have a er 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

Page 70: Lesson 13: Exponential and Logarithmic Functions (slides)

The magic numberDefini on

e = limn→∞

(1+

1n

)n

So now con nuously-compounded interest can be expressed as

B(t) = Pert.

Page 71: Lesson 13: Exponential and Logarithmic Functions (slides)

The magic numberDefini on

e = limn→∞

(1+

1n

)n

So now con nuously-compounded interest can be expressed as

B(t) = Pert.

Page 72: Lesson 13: Exponential and Logarithmic Functions (slides)

Existence of eSee Appendix B

I We can experimentallyverify that this numberexists and is

e ≈ 2.718281828459045 . . .

I e is irra onalI e is transcendental

n(1+

1n

)n

1 22 2.25

3 2.3703710 2.59374100 2.704811000 2.71692106 2.71828

Page 73: Lesson 13: Exponential and Logarithmic Functions (slides)

Existence of eSee Appendix B

I We can experimentallyverify that this numberexists and is

e ≈ 2.718281828459045 . . .

I e is irra onalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.37037

10 2.59374100 2.704811000 2.71692106 2.71828

Page 74: Lesson 13: Exponential and Logarithmic Functions (slides)

Existence of eSee Appendix B

I We can experimentallyverify that this numberexists and is

e ≈ 2.718281828459045 . . .

I e is irra onalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374

100 2.704811000 2.71692106 2.71828

Page 75: Lesson 13: Exponential and Logarithmic Functions (slides)

Existence of eSee Appendix B

I We can experimentallyverify that this numberexists and is

e ≈ 2.718281828459045 . . .

I e is irra onalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.70481

1000 2.71692106 2.71828

Page 76: Lesson 13: Exponential and Logarithmic Functions (slides)

Existence of eSee Appendix B

I We can experimentallyverify that this numberexists and is

e ≈ 2.718281828459045 . . .

I e is irra onalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692

106 2.71828

Page 77: Lesson 13: Exponential and Logarithmic Functions (slides)

Existence of eSee Appendix B

I We can experimentallyverify that this numberexists and is

e ≈ 2.718281828459045 . . .

I e is irra onalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828

Page 78: Lesson 13: Exponential and Logarithmic Functions (slides)

Existence of eSee Appendix B

I We can experimentallyverify that this numberexists and is

e ≈ 2.718281828459045 . . .

I e is irra onalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828

Page 79: Lesson 13: Exponential and Logarithmic Functions (slides)

Existence of eSee Appendix B

I We can experimentallyverify that this numberexists and is

e ≈ 2.718281828459045 . . .

I e is irra onal

I e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828

Page 80: Lesson 13: Exponential and Logarithmic Functions (slides)

Existence of eSee Appendix B

I We can experimentallyverify that this numberexists and is

e ≈ 2.718281828459045 . . .

I e is irra onalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828

Page 81: Lesson 13: Exponential and Logarithmic Functions (slides)

Meet the Mathematician: Leonhard EulerI Born in Switzerland, livedin Prussia (Germany) andRussia

I Eyesight trouble all hislife, blind from 1766onward

I Hundreds ofcontribu ons to calculus,number theory, graphtheory, fluid mechanics,op cs, and astronomy

Leonhard Paul EulerSwiss, 1707–1783

Page 82: Lesson 13: Exponential and Logarithmic Functions (slides)

A limitQues on

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

Page 83: Lesson 13: Exponential and Logarithmic Functions (slides)

A limitQues on

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

Page 84: Lesson 13: Exponential and Logarithmic Functions (slides)

A limit

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

Page 85: Lesson 13: Exponential and Logarithmic Functions (slides)

OutlineDefini on of exponen al func ons

Proper es of exponen al Func ons

The number e and the natural exponen al func onCompound InterestThe number eA limit

Logarithmic Func ons

Page 86: Lesson 13: Exponential and Logarithmic Functions (slides)

Logarithms

Defini on

I The base a logarithm loga x is the inverse of the func on ax

y = loga x ⇐⇒ x = ay

I The natural logarithm ln x is the inverse of ex. Soy = ln x ⇐⇒ x = ey.

Page 87: Lesson 13: Exponential and Logarithmic Functions (slides)

Facts about Logarithms

Facts

(i) loga(x1 · x2) = loga x1 + loga x2

(ii) loga

(x1x2

)= loga x1 − loga x2

(iii) loga(xr) = r loga x

Page 88: Lesson 13: Exponential and Logarithmic Functions (slides)

Facts about Logarithms

Facts

(i) loga(x1 · x2) = loga x1 + loga x2

(ii) loga

(x1x2

)= loga x1 − loga x2

(iii) loga(xr) = r loga x

Page 89: Lesson 13: Exponential and Logarithmic Functions (slides)

Facts about Logarithms

Facts

(i) loga(x1 · x2) = loga x1 + loga x2

(ii) loga

(x1x2

)= loga x1 − loga x2

(iii) loga(xr) = r loga x

Page 90: Lesson 13: Exponential and Logarithmic Functions (slides)

Logarithms convert products to sumsI 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

Page 91: Lesson 13: Exponential and Logarithmic Functions (slides)

ExamplesExample

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

Solu on

I 2 ln 4− ln 3 = ln 42 − ln 3 = ln42

3

I notln 42

ln 3!

Page 92: Lesson 13: Exponential and Logarithmic Functions (slides)

ExamplesExample

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

Solu on

I 2 ln 4− ln 3 = ln 42 − ln 3 = ln42

3

I notln 42

ln 3!

Page 93: Lesson 13: Exponential and Logarithmic Functions (slides)

ExamplesExample

Write as a single logarithm: ln34+ 4 ln 2

Solu on

ln34+ 4 ln 2 = ln 3− ln 4+ 4 ln 2 = ln 3− 2 ln 2+ 4 ln 2

= ln 3+ 2 ln 2 = ln(3 · 22) = ln 12

Page 94: Lesson 13: Exponential and Logarithmic Functions (slides)

ExamplesExample

Write as a single logarithm: ln34+ 4 ln 2

Solu on

ln34+ 4 ln 2 = ln 3− ln 4+ 4 ln 2 = ln 3− 2 ln 2+ 4 ln 2

= ln 3+ 2 ln 2 = ln(3 · 22) = ln 12

Page 95: Lesson 13: Exponential and Logarithmic Functions (slides)

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

Page 96: Lesson 13: Exponential and Logarithmic Functions (slides)

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

Page 97: Lesson 13: Exponential and Logarithmic Functions (slides)

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

Page 98: Lesson 13: Exponential and Logarithmic Functions (slides)

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

Page 99: Lesson 13: Exponential and Logarithmic Functions (slides)

Change of base formula for logarithmsFact

If 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 aI Therefore

y = loga x =logb xlogb a

Page 100: Lesson 13: Exponential and Logarithmic Functions (slides)

Change of base formula for logarithmsFact

If 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 aI Therefore

y = loga x =logb xlogb a

Page 101: Lesson 13: Exponential and Logarithmic Functions (slides)

Example of changing base

Example

Find log2 8 by using log10 only.

Solu on

log2 8 =log10 8log10 2

≈ 0.903090.30103

= 3

Surprised? No, log2 8 = log2 23 = 3 directly.

Page 102: Lesson 13: Exponential and Logarithmic Functions (slides)

Example of changing base

Example

Find log2 8 by using log10 only.

Solu on

log2 8 =log10 8log10 2

≈ 0.903090.30103

= 3

Surprised? No, log2 8 = log2 23 = 3 directly.

Page 103: Lesson 13: Exponential and Logarithmic Functions (slides)

Example of changing base

Example

Find log2 8 by using log10 only.

Solu on

log2 8 =log10 8log10 2

≈ 0.903090.30103

= 3

Surprised?

No, log2 8 = log2 23 = 3 directly.

Page 104: Lesson 13: Exponential and Logarithmic Functions (slides)

Example of changing base

Example

Find log2 8 by using log10 only.

Solu on

log2 8 =log10 8log10 2

≈ 0.903090.30103

= 3

Surprised? No, log2 8 = log2 23 = 3 directly.

Page 105: Lesson 13: Exponential and Logarithmic Functions (slides)

Upshot of changing baseThe 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 func ons are mul ples of each other. Sojust pick one and call it your favorite.

I Engineers like the common logarithm log = log10I Computer scien sts like the binary logarithm lg = log2I Mathema cians like natural logarithm ln = loge

Naturally, we will follow the mathema cians. Just don’t pronounceit “lawn.”

Page 106: Lesson 13: Exponential and Logarithmic Functions (slides)

..“lawn”

..

Image credit: Selva

Page 107: Lesson 13: Exponential and Logarithmic Functions (slides)

Summary

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