Logarithm and Its Uses Physics Notes
Logarithm and Its Uses:
Logarithms were introduced by John Napier in the early century. It was introduced to simplify calculations. They are used by Navigators, Scientists, Engineers, and others to perform computations easily. The logarithm of a number is the exponent to another fixed value called the base, which must be raised to produce that number. In simple cases, the logarithm counts repeated multiplication.
The logarithm can be calculated for any two positive real numbers a and y, where a is not equal to
1. The logarithm of y to base a, denoted log0 (y), is the unique real number x such that
ax = y
For example, 64= 26,
then, log2(64) = log2(26) = 6
The logarithm to base 10 (i.e., a = 10) is called the common logarithm. The natural logarithm has the number e(≈ 2.718) as its base. The binary logarithm uses base 2 (i.e., a = 2).
How to Find out the Logarithm of Any Number:
There are two parts of the logarithm of the number
- Characteristic
- Mantissa.
The fractional part of a logarithm is usually written as a decimal. The whole number part of a logarithm is called the Characteristic.
This part of the logarithm represents the position of the decimal point in the associated number. The decimal part of a logarithm is called the Mantissa.
The mantissa of a common logarithm is always the same regardless of the position of the decimal point in that number.
For example,
log 5270 = 3.72181
The mantissa is 0.72181 and the characteristic is 3.
(a) To Find out the Characteristic:
It is to be noted that a common logarithm is simply an exponent of base 10. Characteristic is the power of 10 when a number is written in scientific notation.
The characteristic can be determined by using the following rules:
1. For a number greater than 1 (> 1): The characteristic is positive and is one less than the number of digits to the left of the decimal point in the number.
2. For a positive number less than 1(< 1): The characteristic is negative and has an absolute value of one more than the number of zeroes (0s) between the decimal point and the first non-zero digit of the number.
The negative characteristic is shown by placing the – (bar) symbol over the number.
For example, log 0.023 = 2.36173
The characteristic is 2 (as 0.023 = 2.3 × 10-2).
The bar over 2 indicates that only the character is negative. So, the logarithm is – 2 + 0.36173.
(b) To Find out the Mantissa:
The mantissa is the decimal part of a logarithm. The logarithm table usually contains only mantissa.
The mantissa can be determined as follows:
The first column of the logarithm table contains the number and the sixth column contains its logarithm. For example, if we want to find the logarithm of 45, then, we will find out the number 45 in the first column. Its logarithm will be 1.65321 in the logarithm table.
Suppose, we have to find out the logarithm of the number 450, but it does not appear in the logarithm table, then we will find out the number 45 in the first column. Notice that both the numbers ‘45’ and ‘450’ have the same mantissa but different characteristics. So, the logarithm of the number 450 will be 2.65321.
Examples of Logarithms of Some Numbers
Method to Find out the Antilogarithms:
The antilogarithm of the logarithm of a number is the number itself. For example, log 1268 = 3.1031, then the antilog (3.1031) = 1268.
How to Find out the Antilogarithm:
- Separate the characteristic and the mantissa.
- Use the antilog table to find out the corresponding value of the mantissa. Look for the row number consisting of the first two digits of the mantissa. Then find out the column number equal to the third digit of the mantissa.
- Find out the value from the mean difference columns. The antilogarithm table also has a set of columns known as the ‘mean difference column’. Now, look at the same row to find out the column number equal to the fourth digit of the mantissa.
- Add the values from the mean difference columns.
- Insert the decimal point after the number of digits that corresponds to the characteristic plus one (1).