Lesson 13: Exponential and Logarithmic Functions (handout)

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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

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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

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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.

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

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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

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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

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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.

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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

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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).

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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 .

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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 .

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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

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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.

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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

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Graphs of various exponentialfunctions

.. x.

y

.y = 1x

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y = 2x

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y = 3x

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y = 10x

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y = 1.5x

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y = (1/2)x

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y = (1/3)x

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y = (1/10)x

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y = (2/3)x

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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

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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

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Simplifying exponentialexpressions

Example

Simplify: 82/3

Solu on

I 82/3 = 3√

82 = 3√64 = 4

I Or,(

3√8)2

= 22 = 4.

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Simplifying exponentialexpressions

Example

Simplify:√8

21/2

Answer2

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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

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y = 2x

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y = 3x

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y = 10x

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y = 1.5x

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y = (1/2)x

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y = (1/3)x

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y = (1/10)x

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y = (2/3)x

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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.

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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.

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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

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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

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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

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The magic numberDefini on

e = limn→∞

(1+

1n

)n

So now con nuously-compounded interest can be expressed as

B(t) = Pert.

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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

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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

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A limitQues on

What is limh→0

eh − 1h

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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

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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

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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.

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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

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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

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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!

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Examples

Example

Write as a single logarithm: ln34+ 4 ln 2

Answerln 12

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Graphs of logarithmic functions

.. x.

y

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y = 2x

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y = log2 x

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(0, 1)

..(1, 0).

y = 3x

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y = log3 x

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y = 10x

.y = log10 x.

y = ex

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y = ln x

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Change of base formula forlogarithms

Fact

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

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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.

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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.”

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Summary

I Exponen als turn sums into productsI Logarithms turn products into sums

I Slide rule scabbards are wicked cool

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Lesson 13: Exponential and Logarithmic Functions (handout)