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USE OF EXPONENTS (LOGARITHMS)

USE OF EXPONENTS (LOGARITHMS)

We know that in scientific work, there are many occasions when we are confronted with the measurements of very small numbers or very large numbers, for example the mass of an electron is found to be 0.000000000000000000000000000 911 g and the number of atoms in 1 mole of an element is found to be 602300,000, 000,000,000000,000 such numbers arc much more conveniently expressed as multiples of 10. For example the first number may be written as 9.11 x 10 g and the second number as 6.023 x 10

By the same approach 0.0005 becomes 5 x 10-4 and 5000 becomes 5 x 103. Thus all numbers may be expressed as a power of 10 and 102, 103, 105, 10-1, 10-2 etc. are generally called as exponential terms in which the base is 10. The powers to the base 10 are known as exponents. Any other number may also be used as a base but it is always convenient to have 10 as the base.

 

The exponential notation is of immense help in simplifying many types of arithmetical computaters and aids in minimization of errors. For example suppose you wish to divide 0.00016 by 80,000. This may be written as:

0.00016/ 80,000 = 16×10-5/ 8 x 104 = 2 x 10-5 x 10-4 = 2 x 10-9

 

The following, more involved calculation shows more clearly, the usefulness of exponential notation.

 

(0.00042) (560) / (84,000)(0.007) = (4.2×10-4)(5.6 x 102) / (8.4×104) (7×10-3)

 

= (4.2 x 5.6) / (8.4 x 7) x (10-4 x 102) / (104 x 10-3)

 

= 0.4 x 10-2/101 = 0.4 x 10-3 = 4.0 x 10-4

 

Setting up calculation in this manner is very convenient and easy.

 

In the expression ax = y, x is called the logarithm of y to the base a, where a must be positive number other than one. A logarithm is, therefore, an exponent and as such, follows the rules applying to exponents.

 

In logarithm the base is usually 10.

 

Suppose N = 10*, now if N = l, 000, then x= 3i.e. 1000= 10’. The power(or exponent) x is called logarithm of the number N. This may be written in algebraic form.

 

Log N = x

 

This is read: “the logarithm of N is x”

The logarithm is divided into two parts, the integer part called the characteristic and the decimal fraction called the mantissa. It must be remembered that the mantissa of a logarithm is always positive; the characteristic may be either positive or negative. Characteristic may be determined by just looking at the numbers. Mantissa may be found out with the help of logarithm tables.

For example, the numbers 0.0025, 0.25, 2.5, 25,000 etc. have the same mantissa but different characteristics.

 

To illustrate how to find the characteristic of a number it is simpler to just consider only numbers of the power of 10.

N Characteristic
1000 (103) 3
100(102) 2
10(101) 1
1(10°) 0
0.1 (10-1) -1
0.001 (l0-3) -3

 

 

Note that characteristic is simply the exponent of the number written as a power of 10. Each of the above numbers has the mantissa 0.000 … as given in the logarithm table. Thus,

 

log 1000 = 3.000

log 100 = 2.000

log 10 = 1.000

log 1 = 0.000

log 0.1 = -1.000

log 0.001 = -3.000

 

Now consider a number 273. The value of 273 will lie between 100 and 1000 i.e. between 102 to 103 which means 273 = l0x where x is between 2 and 3. According to Logarithm table the value of x is equal to 2.4362. It means that 102.4362 = 273 so log 273 = 2.4362

(Characteristic 273 = 2

Mantissa 273 = 0.4362)

THE USE OF LOGARITHMS IN COMPUTATIONS

There are three fundamental rules in logarithms which help us in many calculations in multiplication, division and obtaining roots and powers.

Log ab = log a + log b

Log a/b = log a – log b

Log (a)n = n log a

 

Where a and b are any two positive numbers and n is any positive or negative number.

 

Examples:

(i) Log (450 x 566) = log 450 + log 566

= 2.6532 + 2.7528

= 5.4060

 

Antilog of 0.4060 = 25470

Since the characteristic is 5

Antilog 0.4060 = 254200

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