mathematics - Logarithm Mathematics, Characteristic of Logarithm, Math

Logarithm: Logarithm was invented by John Napier in 1614. 'Logarithm' consists of two parts:

(i)      Log, meaning rule or plan, and

(ii)     Arithm, is the short name of arithmetic.

Logarithm: Thus if a^x = N, then log_a N = × and also if log_a N = x,

Then a^X = N.

First is called the log form and the second is called the exponential form or index form.

Some important deductions:

(i)        a⁰ = 1. for all a > 0

\       log 1 = 0

Thus the logarithm of 1 to any base is zero.

(ii)        a = a \ log_a a = 1

Thus the logarithm of any number to the same base is unity.

LAWS OF LOGARITHMS

1.       Product Rule:

log_x AB = log. A + log_x B

2.       Quotient Rule:

          log_x (A/B) = log_x A - log_xB

3.       Power Rule:

log_x A^B = B log_x A

4.       Change of Base Rule:

log_x A = log_x A / log_x B

          log_B A. log_c B. log₀ C = log₀ A

log_B A. log_A B = 1

[.: log_A A = 1]

or log_B A can be written as 1/ log_AB

5.       Logarithm of a root of a number is the logarithm of the number divided by the index of the root.

(6)     The logarithm of 1 to any base is zero i.e. log. 1 = 0.

(7)     The logarithm of any quantity to the same base is unity, i.e., log. a = 1

(8)     Logarithms to the base 10 are called common logarithms.

(9)     If no base is indicated the base is understood to be 10. Hence log 10 = 1, log 100 = 2, log 1000 = 3, etc.

LOGARITHMS USING FOUR FIGURE TABLES ONLY

Common Logarithms: Logarithms calculated to the base 10 are called common logarithms. Logarithm consists of two parts:

1.       Characteristic [the integral or whole number part]

2.       Mantissa [the fractional or decimal part]

Characteristic: 

1.       To find the characteristic of a number greater than one. "Characteristic is one less than the number of digits to the left of the decimal point in the given number."

2.       To find the characteristic of a number less than one. "Characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and is negative.

Mantissa: To find the mantissa of a number we use logarithmic tables.

1.       The mantissa is the same for the logarithms of all numbers having the same significant digits.

2.       The logarithm of one digit number, say 5, is to be seen in the table, opposite to 50.

3.       The mantissa is always taken positive.

Antilogarithm: If log x = y, then x = antilog of y. i.e.. The number corresponding to a given logarithm is called antilogarithm.

(1)     If the characteristic of the logarithm is positive, then: "Insert the decimal point after (n + 1)th digit, where n = characteristic".

(2)     If the characteristic of the logarithm is negative, then: "Insert the decimal point so that the first significant digit is at 'n'th place, where n = characteristic." 

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