lesson 13: exponential and logarithmic functions (section 021 handout)

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 it backin 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 next week.

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 thanone standard deviation above average.

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


Notes

Notes

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1

Sections 3.1–3.2 : Exponential FunctionsV63.0121.021, Calculus I October 21, 2010

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, includingthe change of base formula.


Outline

Definition of exponential functions

Properties of exponential Functions

The number e and the natural exponential functionCompound InterestThe number eA limit

Logarithmic Functions


Derivation of exponential functions

Definition

If 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 = 8

I 34 = 3 · 3 · 3 · 3 = 81

I (−1)5 = (−1)(−1)(−1)(−1)(−1) = −1


Notes

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2


Anatomy of a power

Definition

A power is an expression of the form ab.

I The number a is called the base.

I The number b is called the exponent.


Fact

If 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


Let’s be conventional

I The desire that these properties remain true gives us conventions forax when x is not a positive whole number.

I For example, what should a0 be? We cannot write down zero a’s andmultiply 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.)

Definition

If a 6= 0, we define a0 = 1.

I Notice 00 remains undefined (as a limit form, it’s indeterminate).


Notes

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3


Conventions for negative exponents

If n ≥ 0, we want

an+(−n) != an · a−n =⇒ a−n

!=

a0

an=

1

an

Definition

If n is a positive integer, we define a−n =1

an.

Fact

I The convention that a−n =1

an“works” for negative n as well.

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

an.


Conventions for fractional exponents

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 iseven.

Notice thatq√

ap =(

q√

a)p

. So we can unambiguously say

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


Conventions for irrational exponents

I So ax is well-defined if a is positive and x is rational.

I What about irrational powers?

Definition

Let a > 0. Thenax = lim

r→xr rational

ar

In other words, to approximate ax for irrational x , take r close to x butrational and compute ar .


Notes

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4


Approximating a power with an irrational exponent

r 2r

3 23 = 8

3.1 231/10 =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


Graphs of various exponential functions

x

y

y = 1x

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


Outline






Notes

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5



Theorem

If a > 0 and a 6= 1, then f (x) = ax 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

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


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

Answer

2


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 and

limx→−∞

ax =∞ x

y

y = 1x

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


Notes

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6


Outline






Compounded Interest

Question

Suppose you save $100 at 10% annual interest, with interest compoundedonce 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 = $121

I $100(1.1)t .


Compounded Interest: quarterly

Question

Suppose you save $100 at 10% annual interest, with interest compoundedfour 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 .


Notes

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7


Compounded Interest: monthly

Question

Suppose you save $100 at 10% annual interest, with interest compoundedtwelve times a year. How much do you have after t years?

Answer

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


Compounded Interest: general

Question

Suppose you save P at interest rate r , with interest compounded n times ayear. How much do you have after t years?

Answer

B(t) = P(

1 +r

n

)nt


Compounded Interest: continuous

Question

Suppose 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 +r

n

)nt= lim

n→∞P

(1 +

1

n

)rnt

= P

[limn→∞

(1 +

1

n

)n

︸︷︷︸independent of P, r , or t

]rt


Notes

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8


The magic number

Definition

e = limn→∞

(1 +

1

n

)n

So now continuously-compounded interest can be expressed as

B(t) = Pert .


Existence of eSee Appendix B

I We can experimentally verifythat this number exists andis

e ≈ 2.718281828459045 . . .

I e is irrational

I e is transcendental

n

(1 +

1

n

)n

1 2

2 2.25

3 2.37037

10 2.59374

100 2.70481

1000 2.71692

106 2.71828


Meet the Mathematician: Leonhard Euler

I Born in Switzerland, lived inPrussia (Germany) andRussia

I Eyesight trouble all his life,blind from 1766 onward

I Hundreds of contributions tocalculus, number theory,graph theory, fluidmechanics, optics, andastronomy

Leonhard Paul EulerSwiss, 1707–1783


Notes

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9


A limit

Question

What is limh→0

eh − 1

h?

Answer

I e = limn→∞ (1 + 1/n)n = lim

h→0(1 + h)1/h. So for a small h, e ≈ (1 + h)1/h.

So

eh − 1

h≈[(1 + h)1/h

]h − 1

h= 1

I It follows that limh→0

eh − 1

h= 1.

I This can be used to characterize e: limh→0

2h − 1

h= 0.693 · · · < 1 and

limh→0

3h − 1

h= 1.099 · · · > 1


Outline






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(x r ) = r loga x


Notes

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Notes

10


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


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: ln3

4+ 4 ln 2

Answer

ln 12


Graphs of logarithmic functions

x

yy = 2x

y = log2 x

(0, 1)

(1, 0)

y = 3x

y = log3 x

y = 10x

y = log10 x

y = ex

y = ln x


Notes

Notes

Notes

11


Change of base formula for exponentials

Fact

If a > 0 and a 6= 1, then

loga x =ln x

ln a

Proof.

I If y = loga x , then x = ay

I So ln x = ln(ay ) = y ln a

I Therefore

y = loga x =ln x

ln a


Example of changing base

Example

Find log2 8 by using log10 only.

Solution

log2 8 =log10 8

log10 2≈ 0.90309

0.30103= 3

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


Upshot of changing base

The point of the change of base formula

loga x =logb x

logb 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


Notes

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12


Summary

I Exponentials turn sums into products

I Logarithms turn products into sums

I Slide rule scabbards are wicked cool


Notes

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13


lesson 13: exponential and logarithmic functions (section 021 handout)

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