Logarithms - Part III

4. Antilogarithm
Antilogarithm is the exact opposite of logarithm of a number.

If x = log b, then antilog (x) = b.

Antilog table for base 10 is readily available. Antilog tables are used for determining the inverse value of the mantissa.  From the characteristic, the position of the decimal point can be determined.

Antilog tables consist of rows that go from .00, . 01, up to .99.  The columns have values 0,1, 2, up to 9. Beyond the 10 columns, there is another column which is known as the mean difference. For determining the antilog of the numbers after the decimal point,  a particular row has to be read off and the mean difference has to be added from the table.

Example 1 : Find the antilog of 2.6992.

The number before the decimal point is 2, so the decimal point will be after the first 3 digits.

From the antilog table, read off the row for .69 and column of 9; the number given in the table is 5000. The mean difference in the same row and under the column 2 is 2. To get the inverse of mantissa add 5000 + 2 = 5002.

Now place a decimal point after the first 3 digits and you get the number 500.2

Thus antilog 2.6992 = 500.2

Example 2 : Find the antilog of  1.0913.

The number before the decimal point is 1, the number of zeroes after the decimal point is zero.

From the antilog table, read off the row for .09 and column of 1; the number given in the table is 1233. The mean difference in the same row and under the column 3 is 1. To get the inverse of mantissa add 1233 + 1 = 1234.

Now place a decimal point before the first digit and you get the number 0.1234.

5. Applications
We will now see how logarithms and antilogarithms of numbers are useful for calculations which are complicated or have very large/small numbers.

Example 1 : Find 80.92 * 19.45.

Let  x = 80.92 * 19.45

Use the log function on both the sides.

log x = log  (80.92 * 19.45)

log  (80.92 * 19.45) = log 80.92 + log 19.45 ( from the laws of logarithms)

From the log tables we get log 80.92 = 1.9080, log 19.45 = 1.2889

Thus log  (80.92 * 19.45)  = 1.9080 + 1.2889 = 3.1969

log x  =  3.1969

Now use antilog functions on both the sides.

x = antilog 3.196

From the antilog tables we see that the antilog of 3.1969 is 1573.0.

Example 2 :  Find  (0.00541 * 4.39)
                                    71.35

Let x = (0.00541 * 4.39)
                    71.35  

Take log functions on both the sides.  

log x = log ( (0.00541 * 4.39) ) ñ log (71.25)          ( from the laws of logarithms)

First term on the RHS : log ( (0.00541 * 4.39) )  =  log (0.00541 * 4.39 )^‡

                                                            = 1/2 log (0.00541)  + 1/2 log (4.39)

log (0.00541) = - 2.2668

‡ log (0.00541) =  - 1.1334

log (4.39) =  0.6423

‡ log (4.39) = 0.3212

Thus the first term on the RHS :  -0.8122

The second term on the RHS : log (71.25) = 1.8527

                                                                                        _
Thus log x =  - 2.6649; in terms of bar, this can be written as 3.3351.

Now take the antilog functions on both the sides, we get x = 0.002163.

Summary 
From all the above examples we can appreciate how simple it is to use the log-antilog tables to calculate tedious numbers. Most of the modern day scientific calculators have log and antilog functions. They give a much more accurate values than the log-antilog tables. The reason for this is that the log-antilog tables consider only four digits.

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