Chabot Mathematics §9.3b Base 10 & e Logs Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot College Mathematics 1 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Review § 9.3 MTH 55 Any QUESTIONS About • §9.3 → Introduction to Logarithms Any QUESTIONS About HomeWork • §9.3 → HW-44 Chabot College Mathematics 2 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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 1. log(10) = 1 2. log(1) = 0 3. log(10x) = x 4. 10logx = x Chabot College Mathematics 3 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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 LOG common logarithms is marked Chabot College Mathematics 4 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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 Chabot College Mathematics 5 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Example Calc Common Log Use a scientific calculator to approximate each number to 4 decimals log130 b) log(0.35) a) log 2,356 Use a scientific calculator to find a) log 2,356 3.3722. log130 b) 4.6365. b) log(0.35) Chabot College Mathematics 6 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Example Sound Intensity This function is sometimes I used to calculate d 10log sound intensity I0 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). Chabot College Mathematics 7 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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 Chabot College Mathematics 8 I d 10log , I0 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Example Sound Intensity Solution (cont.) in the dBA eqn substitute 75 for d and 10−12 for I0 and then solve for I I 75 10 log 12 10 I 1012 I 1012 107.5 7.5 log 12 12 10 10 I 4.5 7.5 10 I 10 10 12 Chabot College Mathematics 9 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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 Chabot College Mathematics 10 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Example Sound Intensity Check If the sound intensity is 10−4.5 W/m2 , verify that the decibel reading is 75. 104.5 d 10log 1012 7.5 d 10log10 d 10 7.5 Chabot College Mathematics 11 d 75 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Graph 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 Chabot College Mathematics 12 f x log x 2 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Graph log by Translation Reflect in x-axis y log x 2 Chabot College Mathematics 13 Shift UP 2 units y 2 log x 2 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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? Chabot College Mathematics 14 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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 Chabot College Mathematics 15 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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 Chabot College Mathematics 16 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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 1. ln(e) = 1 2. ln(1) = 0 3. ln(ex) = x 4. elnx = x Chabot College Mathematics 17 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Example Evaluate ln Evaluate each expression 1 4 b. ln 2.5 a. ln e e Solution 4 a. ln e 4 c. ln 3 1 2.5 b. ln 2.5 ln e 2.5 e c. ln 3 1.0986123 Chabot College Mathematics 18 (Use a calculator.) Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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: 1 A t ln r P Chabot College Mathematics 19 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt 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. Chabot College Mathematics 20 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Example Compound Interest Solution 1 100, 000 t ln 0.08 25, 000 1 t ln 4 0.08 t 17.33 Substitute. Divide. Approximate using a calculator. State Answer The account balance will reach $100,000 in about 17.33 years. Chabot College Mathematics 21 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Example Compound Interest Check: 1 A 17.33 ln 0.08 25, 000 A 1.3864 ln 25, 000 1.3864 ln A ln 25, 000 1.3864 ln 25, 000 ln A 11.513 ln A e11.513 A 100, 007.5 A Because 17.33 was not the exact time, $100,007.45 is reasonable for the Chk Chabot College Mathematics 22 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt WhiteBoard Work Problems From §9.3 Exercise Set • 52, 58, 64, 70, 72, 90 Loud Noise Safe Exposure Time Chabot College Mathematics 23 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt All Done for Today “e” to Several Digits Chabot College Mathematics 24 e= 2.7182818284590452353602874713526624 97757247093699959574966967627724076 63035354759457138217852516642742746 63919320030599218174135966290435729 00334295260595630738132328627943490 76323382988075319525101901157383418 79307021540891499348841675092447614 60668082264800168477411853742345442 43710753907774499206955170276183860 62613313845830007520449338265602976 06737113200709328709127443747047230 69697720931014169283681902551510865 746377211125238978442505695369677078 54499699679468644549059879316368892 30098793127736178215424999229576351 48220826989519366803318252886939849 64651058209392398294887933203625094 43117301238197068416140397019837679 32068328237646480429531180232878250 9819455815301756717361332069811250 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt Chabot Mathematics Appendix r s r s r s 2 2 Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu – Chabot College Mathematics 25 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-61_sec_9-3b_Com-n-Nat_Logs.ppt