[email protected] mth55_lec-61_sec_9-3b_com-n-nat_logs.ppt 1 bruce mayer, pe chabot college...

[email protected] • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§9.3b§9.3bBase 10 & Base 10 & ee

LogsLogs



Review §Review §

Any QUESTIONS About• §9.3 → Introduction to Logarithms

Any QUESTIONS About HomeWork• §9.3 → HW-44

9.3 MTH 55



Common LogarithmsCommon 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 ConventionCommon 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



Example Example Calc Common Log Calc Common Log

Use a calculator to approximate each common logarithm. Round to the nearest thousandth if necessary.

a. log(456) b. log(0.00257)

Solution by Calculator LOG key• log(456) ≈ 2.659 → 102.659 = 456

• log(0.00257) ≈ −2.5901 → 10−2.5901 = 0.00257



Example Example Calc Common Log Calc Common Log

Use a scientific calculator to approximate each number to 4 decimals

log130a) log 2,356 b)

log(0.35)

Use a scientific calculator to finda)

b)

log 2,356 3.3722.



Example Example Sound Intensity 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




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



Graph log by TranslationGraph log by Translation

Sketch the graph of y = 2 − log(x − 2) Soln: Graph f(x) = logx and shift Rt 2

units

f x log x f x log x 2



Graph log by TranslationGraph log by Translation Reflect in x-axis

y log x 2

Shift UP 2 units

y2 log x 2



Example Example Total Recall Total Recall

The function P = 95 – 99∙logx models the percent, P, of students who recall the important features of a classroom lecture over time, where x is the number of days that have elapsed since the lecture was given.

What percent of the students recall the important features of a lecture 8 days after it was given?



Example Example Total Recall Total Recall

Solution: Evaluate P = 95 – 99logx when x = 8.

P = 95 – 99log(8)

P = 95 – 99(0.903) [using a calculator]

P = 95 – 89

P = 6

Thus about 6% of the students remember the important features of a lecture 8 days after it is given



Natural LogarithmsNatural 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 LogarithmsNatural 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 Example Evaluate ln Evaluate ln

Evaluate each expression

a. lne4 b. ln1

e2.5 c. ln 3

Solutiona. lne4 4

b. ln1

e2.5 lne 2.5 2.5

(Use a calculator.)c. ln 3 1.0986123



Example Example Compound Interest 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:

1lnA

tr 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 WorkWhiteBoard Work

Problems From §9.3 Exercise Set• 52, 58, 64, 70,

72, 90

Loud NoiseSafe Exposure Time



All Done for TodayAll Done for Today

“e”to Several

Digits

e = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059921817413596629043572900334295260595630738132328627943490763233829880753195251019011573834187930702154089149934884167509244761460668082264800168477411853742345442437107539077744992069551702761838606261331384583000752044933826560297606737113200709328709127443747047230696977209310141692836819025515108657463772111252389784425056953696770785449969967946864454905987931636889230098793127736178215424999229576351482208269895193668033182528869398496465105820939239829488793320362509443117301238197068416140397019837679320683282376464804295311802328782509819455815301756717361332069811250



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

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srsrsr 22

[email protected] mth55_lec-61_sec_9-3b_com-n-nat_logs.ppt 1 bruce mayer, pe chabot college...

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