Logarithm Formula, Inequalities, Indices and Surds

Logarithm Formula, Inequalities, Indices and Surds.

Logarithms, Inequalities, Indices and Surds, partial fractions


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Introduction to logarithm
“The Logarithm of a given number to a given base is the index of the power to which the base must be raised in order to equal the given number.”
It is also known as fundamental logarithmic identity.
Its domain is (0, infinity ) and range is R. a is called the base of the logarithmic function.
When base is ‘e‘ then the logarithmic function is called natural or Napier logarithmic function and when base is 10, then it is called common logarithmic function.

Characteristic and mantissa

Properties and Logarithm Formula

Logarithm inequalities Formula

Indices and Surds

Definition of indices

If a is any non zero real or imaginary number and m is the positive integer, then {a}^m = a.a.a.a……….a (m times). Here a is called the base and mis the index, power or exponent.

Laws of indices

Definition of surds

Any root of a number which can not be exactly found is called a surd.
Order of a surd is indicated by the number denoting the root.
A second order surd is often called a quadratic surd, a surd of third order is called a cubic surd.

Types of surds

(1) Simple surds : A surd consisting of a single term. For example $2sqrt{3},6sqrt{5,}sqrt{5}$etc.
(2) Pure and mixed surds : A surd consisting of wholly of an irrational number is called pure surd.
A surd consisting of the product of a rational  number and an irrational number is called a mixed surd.
(3) Compound surds : An expression consisting of the sum or difference of two or more surds.
(4) Similar surds : If the surds are different multiples of the same surd, they are called similar surds.
(5) Binomial surds : A compound surd consisting of two surds is called a binomial surd.
(6) Binomial quadratic surds : Binomial surds consisting of pure (or simple) surds of order two 

Properties of quadratic surds

(1) The square root of a rational number cannot be expressed as the sum or difference of a rational number and a quadratic surd.
(2) If two quadratic surds cannot be reduced to others, which have not the same irrational part, their product is irrational.
(3) One quadratic surd cannot be equal to the sum or difference of two others, not having the same irrational part.


Rationalization Factors
If two surds be such that their product is rational, then each one of them is called rationalising factor of the other.

Square roots and Cube root of a binomial quadratic surds

Equations involving surds

While solving equations involving surds, usually we have to square, on squaring the domain of the equation extends and we may get some extraneous solutions, and so we must verify the solutions and neglect those which do not satisfy the equation.

Partial Fractions Definition

(1) Proper rational functions 
(2) Improper rational functions 
(3) Partial fractions 

Different cases of partial fractions

(1) When the denominator consists of non-repeated linear factors : 

(2) When the denominator consists of linear factors, some repeated : To each linear factor (xa) occurring r times in the denominator of a proper rational function, there corresponds a sum of r partial fractions.

(3) When the denominator consists of non-repeated quadratic factors 

Partial fractions of improper rational functions

If degree of $f(x)$ is greater than or equal to degree of $g(x)$, then $frac{f(x)}{g(x)}$ is called an improper rational function and every rational function can be transformed to a proper rational function by dividing the numerator by the denominator.
We divide the numerator by denominator until a remainder is obtained which is of lower degree than the denominator.

General method of finding out the constants

(1) Express the given fraction into its partial fractions in accordance with the rules written above.
(2) Then multiply both sides by the denominator of the given fraction and you will get an identity which will hold for all values of x.
(3) Equate the coefficients of like powers of x in the resulting identity and solve the equations so obtained simultaneously to find the various constant is short method. Sometimes, we substitute particular values of the variable x in the identity obtained after clearing of fractions to find some or all the constants. For non-repeated linear factors, the values of x used as those for which the denominator of the corresponding partial fractions become zero.

Important Points to Remember

1. The logarithm of a number is unique i.e., no number can have two different log to a given base.
2.If a  is not rational, $sqrt[n]{a}$ is not a surd.
3. Some times a suitable substitution transforms the given function to a rational fraction which can be integrated by breaking it into partial fractions.

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