[email protected] MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 1 Bruce Mayer, PE Chabot College...

[email protected] • MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§4.2 Log

Functions



Review §

Any QUESTIONS About• §4.1 → Exponential Functions

Any QUESTIONS About HomeWork• §4.1 → HW-18

4.1

http://www.google.com/url?sa=i&rct=j&q=exponential+growth&source=images&cd=&cad=rja&docid=0eoMN7HMHuVz9M&tbnid=nnz2erP5elegqM:&ved=0CAUQjRw&url=http://bensternke.com/2012/04/why-its-hard-to-invest-in-exponential-growth/&ei=dLvlUZHQG8TbigKixIGwDQ&bvm=bv.49405654,d.cGE&psig=AFQjCNFSrFYNnkBDdQlG9wznnOUQlb5nZQ&ust=1374096555260943



§4.2 Learning Goals

Define and explore logarithmic functions and their properties

Use logarithms to solve exponential equations

Examine applications involving logarithms

John Napier (1550-1617) • Logarithm Pioneer

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=92YkaL4XciseMM&tbnid=OsvykDZwpr9ZlM:&ved=0CAgQjRwwAA&url=http://en.wikipedia.org/wiki/John_Napier&ei=TLzlUdDDHIXXiALLw4C4Cg&psig=AFQjCNEMLs_7QFX0NTc3MyLPWMAAl-gVHA&ust=1374096844505822



Logarithm → What is it?

Concept: If b > 0 and b ≠ 1, then

y = logbx is equivalent to x = by

Symbolically

x = by y = logbx

The exponent is the logarithm.

The base is the base of the logarithm.



Logarithm Illustrated

Consider the exponential function y = f(x) = 3x. Like all exponential functions, f is one-to-one. Can a formula for the inverse Function,x = g(y) be found?

f −1(x) ≡ the exponent to which we must raise 3 to get x.

y = 3x x = 3y

y ≡ the exponent to which we must raise 3 to get x.

Need



Logarithm Illustrated

Now define a new symbol to replace the words “the exponent to which we must raise 3 to get x”:

log3x, read “the logarithm, base 3, of x,” or “log, base 3, of x,” means “the exponent to which we raise 3 to get x.”

Thus if f(x) = 3x, then f−1(x) = log3x. Note that f−1(9) = log39 = 2, as 2 is the exponent to which we raise 3 to get 9



Example Evaluate Logarithms

Evaluate:a) log381 b) log31 c) log3(1/9)

Solution:a) Think of log381 as the exponent to which we

raise 3 to get 81. The exponent is 4. Thus, since 34 = 81, log381 = 4.

b) ask: “To what exponent do we raise 3 in order to get 1?” That exponent is 0. So, log31 = 0

c) To what exponent do we raise 3 in order to get 1/9? Since 3−2 = 1/9 we have log3(1/9) = −2



The Meaning of logax

For x > 0 and a a positive constant other than 1, logax is the exponent to which a must be raised in order to get x. Thus,

logax = m means am = x or equivalently, logax is that unique

exponent for which

xa xa log



Example Exponential to Log

Write each exponential equation in logarithmic form.

a. 43 64 b. 1

2

4

1

16c. a 2 7

Soln a. 43 64 log4 64 3

b. 1

2

4

1

16 log1 2

1

164

c. a 2 7 loga 7 2



Example Log to Exponential

Write each logarithmic equation in exponential form

a. log3 243 5 b. log2 5 x c. loga N x

Soln a. log3 243 5 243 35

b. log2 5 x 5 2x

c. loga N x N ax




Find the value of each of the following logarithmsa. log5 25 b. log2 16 c. log1 3 9

d. log7 7 e. log6 1 f. log4

1

2 Solution

a. log5 25 y 25 5y or 52 5y y 2

b. log2 16 y 16 2y or 24 2y y 4




Solution (cont.)

d. log7 7 y 7 7y or 71 7y y 1

e. log6 1 y 1 6y or 60 6y y 0

f. log4

1

2y

1

24 y or 2 1 22 y y

1

2

c. log1 3 9 y 9 1

3

y

or 32 3 y y 2



Example Use Log Definition

Solve each equation for x, y or z

a. log5 x 3 b. log3

1

27y

c. logz 1000 3 d. log2 x2 6x 10 1

a. log5 x 3

x 5 3

x 1

53 1

125

The solution set is 1

125

.

Solution



Example Use Log Definition

Solution (cont.)

b. log3

1

27y

1

273y

3 3 3y

3 y

c. logz 1000 3

1000 z3

103 z3

10 z

The solution set is 3 .

The solution set is 10 .



Inverse Property of Logarithms

Recall Def: For x > 0, a > 0, and a ≠ 1,

y loga x if and only if x ay .

In other words, The logarithmic function is the inverse function of the exponential function; e.g.

xaxa xxa

a loglog



Derive Change of Base Rule

Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10



Example Inverse Property

Evaluate: 5log 235 . Solution

Remember that log523 is the exponent to which 5 is raised to get 23. Raising 5 to that exponent, obtain

5log 235 23.



Basic Properties of Logarithms

For any base a > 0, with a ≠ 1, Discern from the Log Definition

1. Logaa = 1• As 1 is the exponent to which a

must be raised to obtain a (a1 = a)

2. Loga1 = 0• As 0 is the exponent to which a

must be raised to obtain 1 (a0 = 1)



Graph Logarithmic Function

Sketch the graph of y = log3x

Soln:MakeT-Table→

x y = log3x (x, y)

3–3 = 1/27 –3 (1/27, –3)

3–2 = 1/9 –2 (1/9, –2)

3–3 = 1/3 –1 (1/3, –1)

30 = 1 0 (1, 0)

31 = 3 1 (3, 1)

32 = 9 2 (9, 2)



Graph Logarithmic Function

Plot the ordered pairs and connect the dots with a smooth curve to obtain the graph of y = log3x



Example Graph by Inverse Graph y = f(x) = 3x

Solution:Use Inverse Relationfor Logs & Exponentials

Reflect the graph of y = 3x across the line y = x to obtain the graph of y = f−1(x) = log3x



Properties of Exponential and Logarithmic Functions

Exponential Function f (x) = ax

Logarithmic Function f (x) = loga x

Domain (0, ∞) Range (–∞, ∞)

Domain (–∞, ∞) Range (0, ∞)

x-intercept is 1 No y-intercept

y-intercept is 1 No x-intercept

x-axis (y = 0) is the horizontal asymptote

y-axis (x = 0) is the vertical asymptote



Properties of Exponential and Logarithmic Functions

Exponential Function f (x) = ax

Logarithmic Function f (x) = loga x

Is one-to-one, that is, logau = logav if and only if u = v

Is one-to-one , that is, au = av if and only if u = v

Increasing if a > 1 Decreasing if 0 < a < 1

Increasing if a > 1 Decreasing if 0 < a < 1



Graphs of Logarithmic Fcns

f (x) = loga x (0 < a < 1)f (x) = loga x (a > 1)



Common Logarithms

The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: logx = log10x. So

y = logx if and only if x = 10y

Applying the basic properties of logs1. log(10) = 1

2. log(1) = 0

3. log(10x) = x

4. 10logx = x



Common Log Convention

By this Mathematics CONVENTION the abbreviation log, with no base written, is understood to mean logarithm base 10, or a common logarithm. Thus,

log21 = log1021

On most calculators, the key for common logarithms is marked

LOG



Natural Logarithms

Logarithms to the base “e” are called natural logarithms, or Napierian logarithms, in honor of John Napier, who first “discovered” logarithms.

The abbreviation “ln” is generally used with natural logarithms. Thus,

ln 21 = loge 21.

On most calculators, the key for natural logarithms is marked LN



Natural Logarithms

The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = loge x. So

y = lnx if and only if x = ey

Applying the basic properties of logs1. ln(e) = 1

2. ln(1) = 0

3. ln(ex) = x

4. elnx = x



Example Sound Intensity

This function is sometimes used to calculate sound intensity

010log

Id

I

Where

• d ≡ the intensity in decibels, • I ≡ the intensity watts per unit of area• I0 ≡ the faintest audible sound to the

average human ear, which is 10−12 watts per square meter (1x10−12 W/m2).




Use the Sound Intensity Equation (a.k.a. the “dBA” Eqn) to find the intensity level of sounds at a decibel level of 75 dB?

Solution: We need to isolate the intensity, I, in the dBA eqn 0

10log ,I

dI




Solution (cont.) in the dBA eqn substitute 75 for d and 10−12 for I0 and then solve for I

1275 10 log

10

I

127.5 log

10

I

7.5

1210

10

I

12 12 7.512

10 10 1010

I

4.510 I




Thus the Sound Intensity at 75 dB is 10−4.5 W/m2 = 10−9/2 W/m2

Using a Scientific calculator and find that I = 3.162x10−5 W/m2 = 31.6 µW/m2




CheckIf the sound intensity is 10−4.5 W/m2 , verify that the decibel reading is 75.

4.5

1210

10log10

d

7.510log10d

10 7.5d

75d



Summary of Log Rules

For any positive numbers M, N, and a with a ≠ 1, and whole number p

log log log ;a a aM

M NN

log log ;pa aM p M

log .ka a k

log ( ) log log ;a a aMN M N Product Rule

Power Rule

Quotient Rule

Base-to-Power Rule



Typical Log-Confusion

Beware that Logs do NOT behave Algebraically. In General:

loglog ,

loga

aa

MM

N N

log ( ) (log )(log ),a a aMN M N

log ( ) log log ,a a aM N M N

log ( ) log log .a a aM N M N



Change of Base Rule

Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then logbx can be converted to a different base as follows:

logb x loga x

loga b

log x

logb

ln x

lnb

(base a) (base 10) (base e)



Derive Change of Base Rule

Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10



Example Evaluate Logs

Compute log513 by changing to (a) common logarithms (b) natural logarithms

Soln

b. log5 13 ln13

ln 51.59369

a. log5 13 log13

log 5

1.59369



Use the change-of-base formula to calculate log512. • Round the answer to four decimal places

Solution


5

log12log 12

log5

1.5440

1.54405 12.0009 12 Check



Find log37 using the change-of-base formula

Solution


Substituting into log

log .log

ab

a

MM

b

0.84509804

0.47712125

1.7712

103

10

log 7log 7

log 3

000.73 7712.1



Example Use The Rules

Express as an equivalent expression using individual logarithms of x, y, & z

Solna)

334 7

a) log b) logbx xy

yz z

= log4x3 – log4 yz

= 3log4x – log4 yz

= 3log4x – (log4 y + log4z)

= 3log4x – log4 y – log4z

3

4a) log x

yz



Example Use The Rules

Solnb)

1/ 33

7 7 b) log logb b

xy xy

z z

71

log3 b

xy

z

71log log

3 b bxy z

1log log 7log

3 b b bx y z



Caveat on Log Rules

Because the product and quotient rules replace one term with two, it is often best to use the rules within parentheses, as in the previous example

1log log 7log

3 b b bx y z

71

log3 b

xy

z



Example Cesium-137 ½-Life

A sample of radioactive Cesium-137 has been Stored, unused, for cancer treatment for 2.2 years. In that time, 5% of the original sample has decayed.

What is the half-life (time required to reduce the radioactive substance to half of its starting quantity) of Cesium-137?

http://www.google.com/url?sa=i&rct=j&q=cesium&source=images&cd=&cad=rja&docid=wUv7Pk5eTJ85tM&tbnid=zhNtrhI_zl9-3M:&ved=0CAUQjRw&url=http://www.periodictable.com/Elements/055/&ei=z8vlUYG-I8n5iwLuzIGIDg&bvm=bv.49405654,d.cGE&psig=AFQjCNGnQELbVQQ81_4-iOSAandnDxTMCQ&ust=1374100778611304




SOLUTION: Start with the math model

for exponential Decay Recall the Given information: after 2.2

years, 95% of the sample remains Use the Model and given data to find k Use data in Model: Divide both sides by A0:

A t A0ekt

2.20095.0 keAA

ke 2.295.0




Now take the ln of both Sides

Using the Base-to-Power Rule

Find by Algebra

Now set the amount, A, to ½ of A0

kk ee 2.22.2 ln95.0ln95.0ln

kee ke

k 2.295.0lnlogln95.0ln 2.22.2

HLtHL eA

AtA 0233.0

00

2




After dividing both sides by A0

Taking the ln of Both Sides

Solving for the HalfLife

State: The HalfLife of Cesion-137 is approximately 29.7 years

HLte 0233.05.0

HLt te HL 0233.06931.0ln5.0ln 0233.0

73.290233.0

6931.0

HLt



Example Compound Interest

In a Bank Account that Compounds CONTINUOUSLY the relationship between the $-Principal, P, deposited, the Interest rate, r, the Compounding time-period, t, and the $-Amount, A, in the Account:

1ln

At

r P




If an account pays 8% annual interest, compounded continuously, how long will it take a deposit of $25,000 to produce an account balance of $100,000?

FamiliarizeIn the Compounding Eqn replace P with 25,000, r with 0.08, A with $100,000, and then simplify.




Solution

17.33t

Substitute.

Divide.

Approximate using a calculator.

1 100,000ln

0.08 25,000t

1ln 4

0.08t

State AnswerThe account balance will reach $100,000 in about 17.33 years.




Check: 1

17.33 ln0.08 25,000

A

1.3864 ln25,000

A

1.3864 ln ln 25,000A 1.3864 ln 25,000 ln A

11.513 ln A11.513e A

100,007.5 A

Because 17.33 was not the exact time, $100,007.45 is reasonable for the Chk



WhiteBoard Work

Problems From §4.2• P72 → Atmospheric

Pressure at Altitude– See also: B. Mayer,

“Small Signal Analysis of Source Vapor Control Requirements for APCVD”, IEEE Transactions on Semiconductor Manufacturing, vol. 9, no. 3, 1996, pg 355

http://www.google.com/url?sa=i&rct=j&q=mountain%20climbing%20oxygen%20mask&source=images&cd=&cad=rja&docid=W-l8KBdK5ujXYM&tbnid=m2NZgj-GzdAxtM:&ved=0CAUQjRw&url=http://www.allposters.com/-sp/Mountain-Climbing-at-High-Altitude-Using-Oxygen-Masks-with-the-Cylinders-Strapped-to-Their-Backs-Posters_i1864916_.htm&ei=Vd_lUaDGNoWtigLrrYGIDA&bvm=bv.49405654,d.cGE&psig=AFQjCNEwiFBeG5fnfrIZK6Beek_e8pt7_g&ust=1374105762685110



All Done for Today

Napier’sMasterWorkYear 1619

http://www.google.com/url?sa=i&rct=j&q=john%20napier&source=images&cd=&cad=rja&docid=eqUbfAnEfzwWIM&tbnid=rsTgmh38HFhO-M:&ved=0CAUQjRw&url=http://www.giovannipastore.it/John_Napier.htm&ei=sbzlUciaDcG1iwK_4IHIBg&psig=AFQjCNE0wM176rHTn7UyvnlrDijgsJ18Gw&ust=1374096814029128



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



ConCavity Sign Chart

a b c

−−−−−−++++++ −−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign

Critical (Break)Points Inflection NO

InflectionInflection







http://www.google.com/url?sa=i&rct=j&q=derivative+power+rule&source=images&cd=&cad=rja&docid=NjbfKj_JZBSzwM&tbnid=jJkxz7X095T-NM:&ved=0CAUQjRw&url=http://mathworld.wolfram.com/Derivative.html&ei=TfPWUe_nA6akiQKZ64DIDw&bvm=bv.48705608,d.cGE&psig=AFQjCNGw-AM0Kc2tARyUGs-zPM7iK-qH3g&ust=1373127702659906

http://www.google.com/url?sa=i&rct=j&q=related+rates+calculus&source=images&cd=&cad=rja&docid=wSGtOAIrRxIoIM&tbnid=l4Q1f5xFVNRraM:&ved=0CAUQjRw&url=http://www.cantonschools.org/~lforastiere/ap%20calculus?Templates=Printable&ei=2TLcUY-gOa2UjALp4IHgDw&bvm=bv.48705608,d.cGE&psig=AFQjCNH42r6LfHpjrPyJXK2DJrvdwHWZIQ&ust=1373471717102765

http://www.google.com/url?sa=i&rct=j&q=related+rates+calculus&source=images&cd=&cad=rja&docid=nuuUeFKHM5pqjM&tbnid=x7raEhpCGJ8edM:&ved=0CAUQjRw&url=http://designatedderiver.wikispaces.com/Games+and+fun+stuff&ei=IzPcUYPxBaOLiwLjs4GYAQ&bvm=bv.48705608,d.cGE&psig=AFQjCNH42r6LfHpjrPyJXK2DJrvdwHWZIQ&ust=1373471717102765

Date post:	17-Dec-2015
Category:	Documents
Upload:	louise-rose
View:	220 times
Download:	1 times

[email protected] MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 1 Bruce Mayer, PE Chabot College...

Documents