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Logarithm

  1. Logarithm: Ifais a positive real number, other than 1 andam=x, then we write:
    m= logaxand we say that the value of logxto the baseaism. Examples: (i). 1031000 log101000 = 3. (ii). 34= 81 log381 = 4.
    (iii). 2-3= 1 log2 1 = -3.
    8 8
    (iv). (.1)2= .01 log(.1).01 = 2.
  2. Properties of Logarithms: 1. loga(xy) = logax+ logay
    2. loga x = logax- logay
    y
    3. logxx= 1 4. loga1 = 0 5. loga(xn) =n(logax)
    6. logax= 1
    logxa
    7. logax= logbx = logx .
    logba loga
  3. Common Logarithms: Logarithms to the base 10 are known as common logarithms.
  4. The logarithm of a number contains two parts, namely 'characteristic' and 'mantissa'. Characteristic:The internal part of the logarithm of a number is called itscharacteristic. Case I: When the number is greater than 1. In this case, the characteristic is one less than the number of digits in the left of the decimal point in the given number. Case II: When the number is less than 1. In this case, the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative. Instead of -1, -2 etc. we write1(one bar),2(two bar), etc. Examples:-
    Number Characteristic Number Characteristic
    654.24 2 0.6453 1
    26.649 1 0.06134 2
    8.3547 0 0.00123 3
    Mantissa: The decimal part of the logarithm of a number is known is itsmantissa.For mantissa, we look through log table.



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