LOGARITHMS: BASIC TO ADVANCE FOR CAT SHRUTI MEHRA

INTRODUCTION Logarithms and indices are closely related, and in order to understand logarithms a good knowledge of indices is required. We know that 16 24 Here, the number 4 is the power/exponent/index. In the expression 24, the number 2 is called the base. Example: 64- 82. In this example 2 is the power, or exponent, or index. The number 8 is the base.

DEFINITION Let a, b be positive real numbers then ax b can be written as: a>0,b>0,(a is not equal to 1) Here a is the base of logarithm. Note: The base of logarithm can never be equal to 1. i.e. logix is undefined for all x. Example: If we write down that 64 82 then the equivalent statement using logarithms is log 642. If we write down that log, 27 = 3 then the equivalent statement using powers is 3327.

NOTE Consider the expression 16 24. Remember that 2 is the base, and 4 is the power. An alternative, yet equivalent, way of writing this expression is log216- 4. This is stated as 'log to base 2 of 16 equals 4'. We see that the logarithm is the same as the power or index in the original expression. It is the base in the original expression which becomes the base of the logarithm. The two statements 16 24 are equivalent statements and log216 4

write the following in its logarithmic form: x=2512. Logarithms are exponent. x-2512 is equivalent to _log25x Base The definition of a logarithm indicates that a logarithm is an exponent. logabx is the logarithmic form of ax-b. a-b is the exponential form of log,b=x .

TYPES OF LOGARITHMS 1. Natural Logarithm: When the base of logarithm is e', it is known as Natural Logarithm. Thus log,N is called Natural Logarithm. Example: loge5, loge72 etc. 2. Common Logarithm: When the base of algorithm is 10. Example: log.0(100) , log10 (25) etc.

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FORMULA: . log(1+x) = [ x-x72 + x93-xYA + ] log( 1-x) =-{ x + x2/2 + x3/3 + x4/4 + Example: Remember: log102 0.3010 ) log103 0.4771

CHARACTERISTICS AND MANTI SSA Characteristic: The integral part of logarithm is known as characteristic. Mantissa :_The decimal part is known as mantissa and is always positive. For example: log 3274 - 3.5150 The integral part is 3, The characteristic is The Decimal part is .5150 i.e. Mantissa is .5150

EXAMPLE: Number 8.3145 74.8120 568.31 0.457 0.0456 0.001324 Characteristics 0

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Shruti Mehra

Pursuing B.Tech (CS) | Qualified JEE Mains| Class XII :(City Topper) 96%| Class X: 10 cgpa| Loves to Code and Teach | PotterHead | 2 years'

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