Mathematics. Session Logarithms Session Objectives

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Transcript of Mathematics. Session Logarithms Session Objectives

  • Session Logarithms

  • Session ObjectivesDefinitionLaws of logarithmsSystem of logarithmsCharacteristic and mantissaHow to find log using log tablesHow to find antilogApplications

  • Base:Any postive real numberother than oneLogarithms DefinitionLog of N to the base a is xNote: log of negatives and zero are not Defined in Reals

  • Illustrative ExampleThe number log27 is(a) Integer(b) Rational (c) Irrational(d) PrimeSolution:Log27 is an Irrational numberWhy?As there is no rational number, 2 to the power of which gives 7

  • Fundamental laws of logarithms

  • Other laws of logarithmsChange of baseWhere a is any other base

  • Illustrative ExampleSolution:

  • Illustrative ExampleSolution :Hence True

  • Illustrative ExampleSolution:If ax = b, by = c, cz = a, then the value of xyz isa) 0b) 1c) 2d) 3

  • Illustrative ExampleSolution:

  • Illustrative ExampleSolution:

  • Illustrative ExampleSolution:If log32, log3(2x-5) and log3(2x-7/2) are in arithmetic progression, then find the value of x2log3(2x-5) = log32 + log3(2x-7/2)log3(2x-5)2 = log32.(2x-7/2)(2x-5)2 = 2.(2x-7/2)22x -12.2x + 32 = 0, put 2x = y, we gety2- 12y + 32 = 0 (y-4)(y-8) = 0 y = 4 or 8 2x=4 or 8 x = 2 or 3Why

  • Illustrative ExampleSolution:a2+4b2 = 12ab (a+2b)2 = 16ab2log(a+2b) = log 16 + log a + log b2log(a+2b) = 4log 2 + log a + log blog(a+2b) = (4log 2 + log a + log b)

  • System of logarithmsCommon logarithm: Base = 10Log10x, also known as Briggs systemNote: if base is not given base is taken as 10Natural logarithm: Base = eLogex, also denoted as lnxWhere e is an irrational number given by

  • Illustrative ExampleSolution:Hence False

  • Characteristic and MantissaStandard form of decimalp is characteristic of nlog(m) is mantissa of nlog(n)=mantissa+characteristic

  • How to find log(n) using log tables1) Step1: Standard form of decimaln = m x 10p , 1 m < 10Note to find log(n) we have to find the mantissa of n i.e. log(m)2) Step2: Significant digitsIdentify 4 digits from left, starting from first non zero digit of m, inserting zeros at the end if required, let it be abcd

  • How to find log(n) using log tablesExample n = m x 10p, p: characteristic, log(m): mantissaLog(n) = p + log(m)

    nStd. formm x 10ppmabcd1234.561.23456x10331.234512340.0001231.23x10-4-41.2312301001x1022110000.100231.0023x10-1-11.00231002

  • How to find log(n) using log tables3) Step3: Select row abSelect row ab from the logarithmic table4) Step4: Select column cLocate number at column c from the row ab, let it be x5) Step5: Select column of mean difference dIf d 0,Locate number at column d of mean difference from the row ab, let it be yWhat if d = 0? Consider y = 0

  • How to find log(n) using log tables6) Step6: Finding mantissa hence log(n)Log(m) = .(x+y)Log(n) = p + Log(m)Summarize:1) Std. Form n = m x 10p2) Significant digits of m: abcd3) Find number at (ab,c), say x, where ab: row, c: col4) Find number at (ab,d), say y, where d: mean diff5) log(n) = p + .(x+y)Never neglect 0s at end or front

  • Illustrative ExampleFind log(1234.56)1) Std. Form n = 1.23456 x 1032) Significant digits of m: 12343) Number at (12,3) = 08994) Number at (12,4) = 145) log(n) = 3 + .(0899+14) = 3 + 0.0913 = 3.0913Note this

    nStd. formm x 10ppmabcd1234.561.23456x10331.23451234

  • Illustrative ExampleFind log(0.000123)1) Std. Form n = 1.23 x 10-42) Significant digits of m: 12303) Number at (12,3) = 08994) As d = 0, y = 0 Note this5) log(n) = -4 + .(0899+0) = -4 + 0.0899 = -3.9101

    nStd. formm x 10ppmabcd0.0001231.23x10-4-41.231230

  • Illustrative ExampleFind log(100)1) Std. Form n = 1 x 1022) Significant digits of m: 10003) Number at (10,0) = 00004) As d = 0, y = 05) log(n) = 2 + .(0000+0) = 2 + 0.0000 = 2

    nStd. formm x 10ppmabcd1001x102211000

  • Illustrative ExampleFind log(0.10023)1) Std. Form n = 1.0023 x 10-12) Significant digits of m: 10023) Number at (10,0) = 00004) Number at (10,2) = 95) log(n) = -1 + .(0000+9) = -1 + 0.0009 = -0.9991

    nStd. formm x 10ppmabcd0.100231.0023x10-1-11.00231002

  • How to find Antilog(n)(1) Step1: Standard form of numberIf n 0, say n = m.abcdFor bar notation subtract 1, add 1 we getFor eg. If n = -1.2718 = -1 0.2718n = -1-0.2718=-2+1-0.2718n = -2+0.7282

  • How to find Antilog(n)2) Step2: Select row abSelect the row ab from the antilog tableSelect row 72 from table3) Step3: Select column c of abSelect the column c of row ab from the antilog table, locate the number there, let it be xNumber at col 8 of row 72 is 5346, x = 5346

  • How to find Antilog(n)4) Step4: Select col. d of mean diff. Select the col d of mean difference of the row ab from the antilog table, let the number there be y, If d = 0, take y as 0Number at col 2 of mean diff. of row 72 is 2, y = 2

  • How to find Antilog(n)5) Step5: Antilog(n)If n = m.abcd i.e. n 0Antilog(n) = .(x+y) x 10m+1 x = 5346y = 2Antilog(n) = .(5346 + 2) x 10-(2-1)= .5348 x 10-1 = 0.05348

  • Illustrative ExampleFind Antilog(3.0913)1) Std. Form n = 3.0913 = m.abcd2) Row 093) Number at (09,1) = 12334) Number at (09,3) = 1Antilog(3.0913) = .(1233+1) x 103+1= 0.1234 x 104 = 1234Solution:

  • Illustrative ExampleFind Antilog(-3.9101)1) Std. Form n = -3.91012) Row 083) Number at (08,9) = 12274) Number at (08,9) = 3 5) Antilog(-3.9101)Solution:n = -3 0.9101 = -4 + 1 0.9101n = -4 + 0.0899= .(1277+3) x 10-(4-1)= 0.1280 x 10-3= 0.0001280

  • Illustrative ExampleFind Antilog (2)1) Std. Form n = 2 = 2.00002) Row 003) Number at (00,0) = 10004) As d = 0, y = 05) Antilog(2) = Antilog(2.0000)Solution:= .(1000+0) x 102+1= 0.1000 x 103= 100

  • Illustrative ExampleFind Antilog(-0.9991)1) Std. Form n = -0.99912) Row 003) Number at (00,0) = 10004) Number at (00,9) = 25) Antilog(-0.9991)Solution:-0.9991 = -1 + 1 0.9991= -1 + 0.0009= .(1000+2) x 10-(1-1)= 0.1002

  • Applications1) Use in Numerical Calculations2) Calculation of Compound Interest3) Calculation of Population Growth4) Calculation of DepreciationNow take logNow take logNow take log

  • Illustrative ExampleSolution:

  • Solution Cont.= 0.2708x = antilog (0.2708)= 0.1865 101= 1.865

  • Illustrative ExampleSolution:Find the compound interest on Rs. 20,000 for 6 years at 10% per annum compounded annually.= 20000 (1.1)6logA = log [20000 (1.1)6]= log 20000 + log (1.1)6= log (2 104) + 6 log (1.1)= log2 + 4 + 6 log (1.1)= 0.301+ 4 + 6 (0.0414)= 4.5494

  • Solution Cont.log A = 4.5494A = antilog (4.5494)= 0.3543 105= 35430Compound interest = 35430 20000 = 15,430

  • Illustrative ExampleSolution:The population of the city is 80000. If the population increases annually at the rate of 7.5%, find the population of the city after 2 years.= 80000 (1.075)2log p2 = log 80000 + 2 log 1.075= log 8 + 4 + 2 log (1.075)= 0.9031 + 4 + 2 (0.0314)= 4.9659

  • Solution Cont.log p2 = 4.9659p2 = antilog (4.9659) = 0.9245 105= 92450

  • Illustrative ExampleSolution:The value of a washing machine depreciates at the rate of 2% per annum. If its present value is Rs6250, what will be its value after 3 years.= 6250 (0.98)3log v2 = log 6250 + 3 log 0.98= log (6.250 103) + 3 log (9.8 101)= log 6.250 + 3 + 3 log (9.8) 3= 0.7959 + 3 (0.9912)

  • Solution Cont.log v2 = 0.7959 + 3 (0.9912)= 3.7695v2 = antilog (3.7695)= 0.5882 104= Rs. 5882

  • Class Exercise - 1Solution :

  • Class Exercise - 2Solution :a2 + b2 = 7aba2 + b2 + 2ab = 9ab(a + b)2 = 9abtaking log both sides we get

  • Class Exercise - 3Solution :logx = 2 log5 = log52 = log25x = 25Similarly y = 8

  • Class Exercise - 4Solution :

  • Class Exercise - 5Solution :

  • Class Exercise - 6Solution :

  • Class Exercise 7 (i)Solution :x logx + y logy + z logz xx.yy.zz = 1

  • Class Exercise 7 (ii)Solution :

  • Solution Cont.Similarly Hence b loga + a logb = c logb + b logc= a logc + c logalogab.ba = logbc cb = logca ac

  • Class Exercise - 8Solution :Find characteristic, mantissa and log of each of the following(i) 67.77(ii) .0087(i) 67.77 = 6.777 101Characteristic = 1Mantissa = log (6.777)= 0.(8306+5)= 0.8311log 67.77 = 1 + 0.8311 = 1.8311

  • Solution Cont.(ii) 0.087 = 8.7 103Characteristic = 3Mantissa = log (8.7) = 0.(9395 + 0)= 0.9395

  • Class Exercise 9SolutionFind the antilogarithm of each of the following(i) 4.5851(ii) 0.7214(i) Antilog(4.5851)= .(3846 + 1) 105= 38470(ii) Antilog(0.7214) = Antilog(1 + 1 0.7214)= .(1897 + 3) 100= 0.19Antilog(1 + 0.2786)

  • Class Exercise - 10SolutionIf a sum of money amounts to Rs. 100900 in 31 years at 25% per annum compound interest, find the sum.logP = log(100900) 31log (1.25)= log (1.009 105) 31log (1.25)= log (1.009) + 5 31 log (1.25)

  • Solution Cont.log P = 1.9998P = Antilog (1.9998)= 0.9995 102= 99.95= 0.0037 + 5 31 (0.0969)= 5.0037 3.0039= 1.9998