1. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §9.3b Base 10 & e Logs

2. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.3 → Introduction to Logarithms  Any QUESTIONS About HomeWork §9.3 → HW-44 9.3 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 3 Bruce Mayer, PE Chabot College Mathematics Common Logarithms  The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: logx = log 10 x. So y = logx if and only if x = 10 y  Applying the basic properties of logs 1.log(10) = 1 2.log(1) = 0 3.log(10 x ) = x 4.10 logx = x

4. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 4 Bruce Mayer, PE Chabot College Mathematics 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 = log 10 21  On most calculators, the key for common logarithms is marked LOG

5. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 5 Bruce Mayer, PE Chabot College Mathematics 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 → 10 2.659 = 456 log(0.00257) ≈ −2.5901 → 10 −2.5901 = 0.00257

6. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Calc Common Log  Use a scientific calculator to approximate each number to 4 decimals  Use a scientific calculator to find a) b)

7. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 7 Bruce Mayer, PE Chabot College Mathematics 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 I 0 ≡ the faintest audible sound to the average human ear, which is 10 −12 watts per square meter (1x10 −12 W/m 2 ).

8. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 8 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Sound Intensity  Solution (cont.) in the dBA eqn substitute 75 for d and 10 −12 for I 0 and then solve for I

10. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Sound Intensity  Thus the Sound Intensity at 75 dB is 10 −4.5 W/m 2 = 10 −9/2 W/m 2  Using a Scientific calculator and find that I = 3.162x10 −5 W/m 2 = 31.6 µW/m 2

11. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Sound Intensity  Check If the sound intensity is 10 −4.5 W/m 2, verify that the decibel reading is 75. 

12. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 12 Bruce Mayer, PE Chabot College Mathematics Graph log by Translation  Sketch the graph of y = 2 − log(x − 2)  Soln: Graph f(x) = logx and shift Rt 2 units

13. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 13 Bruce Mayer, PE Chabot College Mathematics Graph log by Translation  Reflect in x-axis  Shift UP 2 units

14. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 14 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 15 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 16 Bruce Mayer, PE Chabot College Mathematics 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 = log e 21.  On most calculators, the key for natural logarithms is marked LN

17. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 17 Bruce Mayer, PE Chabot College Mathematics Natural Logarithms  The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = log e x. So y = lnx if and only if x = e y  Applying the basic properties of logs 1.ln(e) = 1 2.ln(1) = 0 3.ln(e x ) = x 4.e lnx = x

18. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Evaluate ln  Evaluate each expression  Solution (Use a calculator.)

19. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 19 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 20 Bruce Mayer, PE Chabot College Mathematics 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?  Familiarize In the Compounding Eqn replace P with 25,000, r with 0.08, A with $100,000, and then simplify.

21. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Compound Interest  Solution Substitute. Divide. Approximate using a calculator.  State Answer The account balance will reach $100,000 in about 17.33 years.

22. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Compound Interest  Check:  Because 17.33 was not the exact time, $100,007.45 is reasonable for the Chk

23. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 23 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.3 Exercise Set 52, 58, 64, 70, 72, 90  Loud Noise Safe Exposure Time

24. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 24 Bruce Mayer, PE Chabot College Mathematics All Done for Today “e” to Several Digits e = 2.7182818284590452353602874713526624 97757247093699959574966967627724076 63035354759457138217852516642742746 63919320030599218174135966290435729 00334295260595630738132328627943490 76323382988075319525101901157383418 79307021540891499348841675092447614 60668082264800168477411853742345442 43710753907774499206955170276183860 62613313845830007520449338265602976 06737113200709328709127443747047230 69697720931014169283681902551510865 746377211125238978442505695369677078 54499699679468644549059879316368892 30098793127736178215424999229576351 48220826989519366803318252886939849 64651058209392398294887933203625094 43117301238197068416140397019837679 32068328237646480429531180232878250 9819455815301756717361332069811250

25. BMayer@ChabotCollege.edu MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 25 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –