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Question

How to solve logs using log tables during practical application calculation . Please explain with an example.

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Solution

There are many methods but we are going to use a convenient one.

Note: Here, we are going to find the log of some number with base 10.

Let’s find the log of 5678.34
For convenient, log 5678.34 = 3.7542
3.7542 has two parts :
1) Characteristic (i.e. before decimal point which is 3)
2) Mantissa (i.e. after decimal point which is 7542)
Note : We use log table to find only Mantissa as characteristic can be found very easily by our self.

We know that log (m×n) = log m + log n So, 5678.34 can be written as 56.7834 × 10^2 . So, log 5678.34 = log 56.7834 + log 10^2
Now log 10^2 = 2 (log m^n = n log m).
Now here comes the main part i.e. log 56.7834

Here are the steps -
In the Log Table, look for the row 56. Then move to the right in the same row until you reach to column 7 ( first digit after decimal point). It should have the value 7536 which should be written as 0.7536 (as it is the decimal part). Now, in the same row, move to mean difference table and note down the value under the column 8 ( 56.7834) which should be 6. Now add the mean difference to the value we obtained in column 7 ( i.e. 0.7536 + 0.0006 = 0.7542).

So we obtained the mantissa part i.e. 0.7542.
Now characteristic of log 56.7834 is one less than the number of digits it has (i.e. 2–1=1)
Now Add Characteristic and mantissa. ( 1 + 0.7542 = 1.7542)

To sum up the steps —
log 5678.34 = log 10^2 + log 56.7834
= 2 + 1.7542
=3.7542
So we got log 5678.34 = 3.7542

Lets consider another example : log 1.35

log 1.35 = log 10^-1 + log 13.5
= -1 + 1.1303
= 0.1303
So log 1.35 = 0.1303

As there are lot of steps involved here so I suggest you to practice some more with other numbers.

Thank You.hope it helps..


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