Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §9.3bBase 10 & e Logs Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. MTH 55 9.3 Review § • Any QUESTIONS About • §9.3 → Introduction to Logarithms • Any QUESTIONS About HomeWork • §9.3 → HW-44

3. 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 logs • log(10) = 1 • log(1) = 0 • log(10x) = x • 10logx = x

4. LOG 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

5. Example  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

6. Example  Calc Common Log • Use a scientific calculator to approximate each number to 4 decimals • Use a scientific calculator to find

7. Example  Sound Intensity • This function is sometimes used to calculate sound intensity • 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).

8. Example  Sound Intensity • 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

9. Example  Sound Intensity • Solution (cont.) in the dBA eqn substitute 75 for d and 10−12 for I0 and then solve for I

10. Example  Sound Intensity • 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

11. Example  Sound Intensity • CheckIf the sound intensity is 10−4.5 W/m2, verify that the decibel reading is 75. 

12. Graph log by Translation • Sketch the graph of y = 2 − log(x − 2) • Soln: Graph f(x) = logx and shift Rt 2 units

13. Reflect in x-axis Graph log by Translation • Shift UP 2 units

14. Example  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?

15. Example  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

16. LN 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

17. 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 logs • ln(e) = 1 • ln(1) = 0 • ln(ex) = x • elnx = x

18. (Use a calculator.) Example  Evaluate ln • Evaluate each expression • Solution

19. 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:

20. Example  Compound Interest • 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.

21. Example  Compound Interest • Solution Substitute. Divide. Approximate using a calculator. • State AnswerThe account balance will reach $100,000 in about 17.33 years.

22. Example  Compound Interest • Check: • Because 17.33 was not the exact time, $100,007.45 is reasonable for the Chk

23. WhiteBoard Work • Problems From §9.3 Exercise Set • 52, 58, 64, 70, 72, 90 • Loud NoiseSafe Exposure Time

24. All Done for Today e = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059921817413596629043572900334295260595630738132328627943490763233829880753195251019011573834187930702154089149934884167509244761460668082264800168477411853742345442437107539077744992069551702761838606261331384583000752044933826560297606737113200709328709127443747047230696977209310141692836819025515108657463772111252389784425056953696770785449969967946864454905987931636889230098793127736178215424999229576351482208269895193668033182528869398496465105820939239829488793320362509443117301238197068416140397019837679320683282376464804295311802328782509819455815301756717361332069811250 “e”to SeveralDigits

25. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –