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Use of Logs and Antilogs

Please choose a topic from the list:

  1. Determining Common Log Using A 4-place Log Table
  2. Computations Using Logs
  3. Determining the Antilog
  4. Using a Calculator To Determine Logs and Antilogs
  5. Practical Chemistry Example using Logs and Antilogs

    Use Of Common Logs

    Common Logarithms are powers to the base 10. Natural logs are exponents to the base "e". "e" = 2.71

    In working with common logs the following procedure is followed:

    1. Convert the number whose log is being determined to scientific notation

    2. Determine the log of the principle number in front of the exponential part of the notation using a log table. There are 4 place, 5 place, and 6 place log tables found in most math and science books located usually in the Appendix.

    3. Determine the log of the exponential part. That is easy because the log of the exponential part is simply the exponent of 10. For example:

      log 107 = 7

      log 103 = 3

      log 10-2 = -2

    4. Add the log from step 2 to the log in step 3 for the final log of the number.

    The difficult part is step 2. You have to have a log table to get it. Lets assume that you have a 4-place log table. Here is how you would determine the log. Let's say you wanted to know the log of 4.18. In the log table there will be rows of four digit numbers. Each row will be labeled at the extreme left of the row with a two digit numbers separated by a decimal. Then there will be columns of four digit numbers each column headed by a single digit from 0-9. Each of the four digit numbers in the table has a decimal to the left most digit which does not show but should be placed in there. So if we want the log of 4.18, we would go to the left most end of the rows and go down until you reach the row labeled with 4.1. We would move to the right in this row. Then we would go to the top of the columns and move over to the column labeled with an 8. We would then proceed downward in that column. If you extend from the 4.1 across to the right and extend from the top of the column marked 8 downward until the two extensions intersect, at the point of intersection will be a four digit number actually called the mantissa of the final log determination. To the left of this four digit number will be a decimal or one should be placed there.

    Let me take an example from the beginning:

    Let's determine the log 25,600

    1. Convert number to standard scientific (exponential) notation. That would be

      2.56 X 104

    2. Determine the log of 2.56 as described above. If you do this you will arrive at a four digit number (mantissa) which is .4082

    3. Determine the log of the exponential part:

      log 104 = 4

    4. Add the two together for the final answer:

      4 + .4082 = 4.4082

    Negative Logs

    It is possible to have negative logs as well. For example, what would be the log of

    0.00314.

    1. Convert to standard exponential notation:

      3.14 X 10-3

    2. Determine the log 3.14 from the table:

      log 3.14 = .4969

    3. Determine the log 10-3 = -3

    4. Add the two from step 2 and step 3 together:

      .4969 + (-3) = -2.5031

      Often they leave negative logs in the first form that is

      .4969 -3

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    Calculation Using Logs

    You can calculate with logs in the following manner

    Multiplication of numbers

    To multiply numbers using logs simply:

    1. Convert each number to a log
    2. Add the logs (because when we multiply numbers we add their log forms)

    3. Determine the antilog for the product

    Division of Numbers

    1. Convert the denominator and the numerator into log forms

    2. Subtract the log form of the denominator from the log form of the numerator to get a difference

    3. Determine the antilog of that difference in step 2 for the final quotient

    Raising A Number To A Power

    1. Convert the number to a log form

    2. Multiply the log form by the exponent (power)

    3. Determine the antilog of step 2 for the final answer

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    The last step in all the above computations using logs involves knowing how to determine the antilog of a logarithm. This is easy if the log is positive.

    Determining the Antilog of a positive log form

    1. Separate the whole number part of the log form (called the characteristic) from the decimal part(called the mantissa the four digits with the decimal to the left most position)

    2. Determine the antilog of the decimal part (mantissa). This is done by going back to the four place log table and browse through the mantissas located throughout the table. Find the mantissa that comes closest to the one you are determining. Then placing your finger on that mantissa extend your finger to the left until it reaches the beginning of the row where the two digit numbers separated by a decimal are. That two digit number is the first two digits in the number. Now take you finger and proceed upward from the found mantissa until you reach the head of the column it was in. That single digit heading that column is the third digit in the antilog.

    3. Determine the antilog of the whole number part of the log form (sometimes called the characteristic). That is the simple part because that would be 10 raised to the power equal to the whole number part. So if the characteristic is 3 then the Antilog of 3 = 103

    4. Multiply the antilog in step 2 with that in step 3 for the final antilog.

    Let's take an example:

    Antilog of 3.8734

    1. Separate the mantissa from the characteristic

      3 and .8734

    2. Determine the antilog of

      .8734.

      If you don't find a mantissa in the table that is .8734 find the one that comes closest. In my four place log table the closest that to .8734 that it has is one that is .8733. Move from there to the left to the front of the row where we find the digits

      7.4

      Moving your finger from the .8733 upward to the top of the column gives the single digit

      7

      So antilog of .8734 = 7.47

    3. Determine the antilog of the characteristic 3 which would be

      Antilog 3 = 103

    4. Multiply the two together for the final antilog

      7.47 X 103

      or 7470

      So the Antilog 3.8734 = 7.47 X 10^3 = 7470

    Determining the Antilog Of A Negative Log

    Determining antilogs of negative log forms can be a bit tricky since a Negative log form has a negative mantissa. All of the mantissas in the 4 place log table are positive mantissas so one has to take the additional step of converting the negative log to a form that has a positive mantissa.

    For example:

    Antilog of -4.5611

    1. Note the characteristic which in this case is a four. Add the next whole number to that log form: 

            -4.5611
      +5
      but if we just add 5 that would change the log so we have to subtract 5 as well:

      -4.5611 +5 -5

      If I do this I will get:

      .4399 - 5

      The above log form is equivalent with the original log form but its mantissa is positive and can now be looked up in the table as we did before.

    2. Determine the antilog of the positive mantissa

      In looking in the table there is no .4399 but the closest mantissa is

      .4393

      so we will take that one and move back to the beginning of that row and also upward to the top of the column it is in and we get:

      2.75

    3. Now we determine the antilog of -5 which would be 10-5

    4. Multiply the two together for the final antilog result:

      2.75 X 10-5

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    Use of A Calculator To Determine Logs and Antilogs

    Of course very few people use log tables anymore. They use a calculator that has a log function key and an antilog function key which is often activated by pressing a second function or inverse function key and then pressing the log key. In other words if you depress the second function key with the log key you would be requesting the calculator to determine the antilog of the inputted value.

    Determining Logs With A Calculator

    Here is how most calculators handle log computations.

    Let's determine the Log 14,500 using a calculator

    1. Input the 14500 by keying in the number on the keyboard

    2. Depress the log key (or ln key for natural log)

    3. Read the display

    Determining An Antilog Using The Calculator

    Let's determine the antilog using a calculator that has a second function key. For example, let's calculate

    Antilog of 3.6783

    1. Input 3.6783 by typing in the numbers including the decimal on to the keyboard of the calculator.

    2. Depress the second function key (or the inv key if your calculator has one)

    3. Depress the log key (or ln key for antiln)

    4. Read the display for the antilog.

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    Practical Chemistry Example Illustrating The Use of Logs and Antilogs On Calculator

    Determining pH From A Given Hydrogen Ion Concentration

    Problem:Calculate the pH of the following solutions for the given H+ concentrations (use your scientific calculator).

    [H+] = 3.2 x 10-5M

    1. Use the pH formula

      pH = -log [H+]

    2. Separate the numerical and exponential parts and determine their logs separately

      pH = -[ log 3.2 + log 10-5]

    3. Determine the log of 3.2 on the calculator. Just input 3.2 then press the "log" key and then press the "=" key sometimes called the "enter" key and then read display.

      Some calculators have you depress the "log" key first and then enter the 3.2 and press the "enter" key.

      Consult the user's manual of your calculator model for further specific instructions

      log 3.2 = 0.505

    4. Determine the log 10-5.

      The log 10x = x

      so the log 10-5 = -5

    5. Add the answer of step 3 to answer for step 4

      [ .505 +(-5) ] = .505 - 5 = -4.495

    6. Multiply the answer of step 5 by a -1 which just changes the sign on step 5. This will be your final answer.

      pH = -[ -4.495] = +4.495

    Determining Hydrogen Ion From pH

    Problem:Determine the Hydrogen Ion Molar concentration of a solution having a pH = 2.2

    1. Use the formula for determining Hydrogen ion from the given pH

      [H+] = Antilog ( -pH )

    2. Insert the pH and the negative sign into formula

      [H]+ = Antilog ( - 2.2 )

    3. Split up the whole number and the decimal part of the 2.2 and determine the Antilogs separately

      [H^+] = Antilog (.2) X Antilog (-2)

    4. Determine the Antilog (.2) on the calculator.

      This is usually done a number of ways but I will describe the most common way. Enter the .2 on the calculator. Press the second function key sometimes called the "inv" key. Then press the "log" key and then press the "=" key or the "enter" key.

      Sometimes a particular brand of calculator have you press the "inv" or second function key Press the "log" key and then key in the .2 then depress the "=" or "enter" key.

      Consult your user's manual for your particular model of calculator

      Antilog (.2) = 1.59

    5. Determine the Antilog of (-2)

      Antilog ( x ) = 10x

      Antilog (-2) = 10-2

    6. Multiply the answer of step 4 and step 5 together for the final answer.

      [H+] = 1.59 X 10-2 Molar

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    R. H. Logan, Instructor of Chemistry, Dallas County Community College District, North Lake College.

    Acknowledgements

    Send Comments to R.H. Logan:

    rhl7460@dcccd.edu 

    All textual content copyrighted (c) 1997
R.H. Logan, Instructor of Chemistry, DCCCD
All Rights reserved

    Revised: 4/22/2002

    URL:http://edie.cprost.sfu.ca/~rhlogan/log.html