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logarithms

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Calculating log of 3976, 3.884, 97.67, 12.8, 12, 51789

  • log10(3976)

    1. Check how many digits are present before decimal and subtract 1 from that number; here there are 4 digits before decimal so we get; 4-1 = 3
    2. We will use this number(3) as our only before decimal number and now we have to figure out what will come after the decimal in our answer
    3. Our original number is a 4 digit number, so in log table we have to check log of first two digits in the third digit (39 in 6; 5988)
    4. Now we check for mean difference of first two digits in the fourth digit (39 in 7; 6)
    5. Now we just add these numbers together and get the numbers after the decimal for our original number. (5988 + 7 = 5995)

    Now we can just put our calculations together to get our answer log10(3976) = 3.5995

  • log10(3.884)

    1. There is 1 digit before decimal so; 1-1 = 0
    2. We have 4 digits in our number. Firstly, in log table we will check the log of 38 in 8; 5888
    3. Now we check mean difference of 38 in 4; 5
    4. We just add these numbers together; 5888 + 5 = 5893

    So, log10(3.884) = 0.5893

  • log10(97.67)

    1. There are two digits before decimal so; 2-1 = 1
    2. Firstly we check log of 97 in 6; 9894
    3. Now we check mean difference of 97 in 7; 3
    4. We just add these numbers together; 9894+3 = 9897

    So, log10(97.67) = 1.9897

  • log10(12.8)

    1. There are two digits before decimal so; 2-1 = 1

    2. We check log of 12 in 8; 1072

      We don’t have to check for mean difference here because the number we have is only a three digit number. We can just find log of first two digits in third digit and we will get the number after the decimal in our answer

    That’s it. So, log10(12.8) = 1.1072

  • log10(12) = log(10)(12.0)

    If the number has less 3 digits then you can just add a 0 after the decimal to increase the digits

    1. There are 2 digits before decimal so; 2-1 = 1
    2. We check log of 12 in 0; 0792

    So, log10(12) = 1.0792

  • log10(51789)

    ⇒ log10(51.789 x 103)

    ⇒ log10(51.789) + log10(103)

    log(m x n) = log(m) + log(n)]

    ⇒ 3 + log10(51.79)

    We can round off 51.789 to be 51.79 && log of 103 is 3

    Now we just have to calculate log10(51.79)

    1. There are 2 digits before decimal so 2-1 = 1
    2. We check log of 51 in 7; 7135
    3. We check mean difference of 51 in 9; 8
    4. We just add these numbers together; 7135+8 = 7143

    So, log10(51.79) = 1.7143

    Hence, log10(51789) = 3 + 1.7143 = 4.7143

Calculating antilog of 3.976, 32.884, 0.7816, 0.051, -2.432, -1.2232

  • antilog(3.976) = antilog(3 + 0.976)

    ⇒ 103 x antilog(0.976)

    antilog(m + n) = antilog(m) x antilog(n) && antilog(10a) = a

    1. During finding antilogarithms, we just have to look at the digits after the decimal, we have 3 digits after the decimal. So, we will look in the antilog table for antilog of first 2 digits in third digit (97 in 6; 9484)
    2. Now we just put a decimal after 1 digit in our antilogarith and we will get our antilog; 9.484

    So, antilog(0.976) = 9.484

    Hence, antilog(3.967) = 9.484 x 103 = 9484

  • antilog(32.884) = antilog(32 + 0.884)

    ⇒ antilog(0.884) x 1032

    1. We check antilog of 88 in 4; 7656
    2. We put a decimal after 1 digit; 7.656

    So, antilog(32.884) = 7.656 x 1032

  • antilog(0.7816)

    1. Firstly we check antilog of 78 in 1; 6039
    2. Now we find antilog of 78 in 6; 8
    3. We just add these numbers together to get our answer; 6047
    4. We just add a decimal after 1 digit; 6.047

    So, antilog(0.7816) = 6.047

  • antilog(0.051)

    1. We just check antilog of 05 in 1; 1125
    2. We place a decimal after 1 digit; 1.125

    So, antilog(0.051) = 1.125

  • antilog(-2.432)

    1. When we have to find antilog of a negative number, we take a whole number just bigger than absolute value of our number; 3

    2. Now we put negative and positive of that number in our antilog; antilog(-3 +3 -2.432)

      We do this so that we can get the inside of our antilog to be a whole number and a positive decimal number smaller than 1

    3. So we have; antilog(-3 + 0.568) ⇒ antilog(0.568) x 10-3

    4. Now we can just for antilog of 56 in 8; 3698

    5. We just put a decimal after 1 digit; 3.698

    So, antilog(0.568) = 3.698

    Hence, antilog(-2.432) = 3.698 x 10-3

  • antilog(-1.2232) = antilog(-2 +2 - 1.2232) ⇒ antilog(0.7768) x 10-2

    1. We just check antilog of 77 in 6; 5970
    2. Now we check mean difference of 77 in 8; 11
    3. We just add these two values together; 5970 + 11 = 5981
    4. Now we put a decimal after 1 digit; 5.981

    So, antilog(0.7768) = 5.981

    Hence, antilog(-1.2232) = 5.981 x 10-2