Logarithms Important Formulas - Aptitude Questions and Answers

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Important Formulas
Logarithms

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General Aptitude
Logarithms
Important Formulas

Definition

Let a be a positive real number other than 1 and a^m = x, then we write m = log_ax and we say that the value of log x to the base a is m.
(e.g) 10³ = 1000 => log₁₀1000 = 3

Properties of Logarithms

Log_amn = log_am + log_an

(i.e) logarithm of the product of two quantities is equal to the sum of their logarithms taken separately.

log_a

= log_am - log_an

(i.e) logarithm of the quotient of two numbers is equal to the difference of their logarithms.

log_a(m)ⁿ = n log_am

(i.e) logarithm of a power of a number is the product of the power and the logarithm of the number.

log_mm = 1

(i.e) logarithm of any number with respect to itself as base is unity.

log_a1 = 0

(i.e) logarithm of unity with respect to any finite quantity (other than zero) as base is zero.

log_ax =

log_xa

Note:

1.	Logarithm to the base 10 are called common logarithms.
2.	When base is not mentioned, it is takes as 10
3.	Logarithm to the base e, is called Napierian logarithm

Characteristic and Mantissa

The integral part of the logarithms of a number is called its characteristics. The decimal part of the logarithms of a number is known as its Mantissa.

(e.g) log 245 = 23892, the characteristic of log 245 is 2 and its mantissa is 0.3892

Note:

1.	The characteristic of the logarithm of a number may be positive or negative but the mantissa is always positive

2.	The inverse operation of the logarithm is called antilogarithm. Thus, if log 245 = 2.3892 => antilog (2.3892) = 245

Rules to find the characteristic and mantissa of the common logarithms:

1.	The characteristic of common logarithm of any number grater than unity is positive and one less than the number of digits in the integral part of the number.

2.	The characteristic of the common logarithm of any positive number less than the unity is negative and is numerically one grater than the number of zero immediately after the decimal point of the number.

3.	The mantissa of the logarithms of all numbers consisting of the same significant digits arranged in the same order but differing only in the position of their decimal points are the same.

Scientific Notation of Numbers:

Sometimes it is very difficult to read and write very big number having lot of digits. It occupies a lot of space even in computer. So a method of representing such number in short form is known as ‘Scientific Notation’

(e.g) The volume of earth = 1083000000000 km³ = 1.083 x10¹² km³

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