Economics: Indices, Logarithm and Equations

Indices

We know that 2×2×2×2=4 factors of 2=2⁴ and a×a×a×a.......to ⁿ factors = aⁿ.
Here, 'n' is called the index of power and is spoken as n^th power of a. The factor which multiplies is called the base i.e. a.

Law of Indices

1. Product Law

The powers of indices of the terms with the same base are added if they are multiplied. For example,

a^m× aⁿ = a^m+n

a^m+n× a^o = a^m+n+o

2. Quotient Law

The index of the quotient of two algebraic terms with the same base is the difference between the indices of the numerator and denominator. For example,

\(\frac{a^m}{a^n}\) = a^m-n = \(\frac{1}{a^{n-m}}\)

\(\frac{a^{m+n}}{a^{2n-m}}\) = a^m+n-2n+m = a^{2m- n}

3. Power Law

The power of an index of any algebraic term is the product of two powers. For example,

(a^m)ⁿ = a^mn

[{(a^p)^q}^r]^s = [{a^pq}^r]^s = [a^pqr]^s = a^pqrs

4. Law of negative index

Any algebraic term with negative index is the reciprocal of the term with the positive index. For example,

a^-m = \(\frac{1}{a^m}\)

\(\frac{1}{a^{-n}}\) = aⁿ

5. Law of zero index

Any algebraic term having the index zero (0) is always equal to 1. But 0⁰ ≠1.

a⁰ = 1

a^m × a^m = a^m-m = a⁰ = 1

6. Root Law

The term with reciprocal power can be expressed as the root of the term. For example,

a^{\(\frac{1}{m}\)} = \(\sqrt[n]{a}\)

It is said as the n^th root of a.

a\(\frac{m}{n}\) = \(\sqrt[n]{m}\)

It is understood that \(\sqrt{a}\) is the square root of a.

7. If the terms with the same bases are equal, then the indices are also equal.

For example, If a^x = a^y , then x = y.

8. If the terms with the same indices are equal, then the bases are also equal.

For example, If x^a = y^a, then x =y.

Logarithms

The extensions of the meanings of indices to all kinds of numbers has great importance in practice because instead of the process of multiplication of the numbers we use that of an addition of the indices. We must have the table of the indices which indicate the power any given number is a selected number such as 10 which we have used. Such a table is called a table of logarithms. And the number such as 10¹, 10^{\(\frac{1}{2}\),} 10^{\(\frac{3}{2}\)} (i.e 10) is the base of logarithm.

If three quantities a, x, N are related by a^x = N.....(i)

Where x is the logarithm of the number to the base a. We write log_aN = x and read as x log of N to the base a.

Antilogarithms

Antilogarithm is the opposite of logarithm. The problem of a logarithm is to find the logarithm of given number while that of antilogarithm is to find the number.
If log_aN = x then N is called antilogarithm of x to the base a.

Equations

An equation shows the relationship between variables and constants connected by the sign of equality.Two equal expressions are called the sides of an equation. Thus, 2x + 5 = 0 is an equation of first degree and is known as a linear equation.

Types of Equation

1. Simple Linear Equation

2. The equationin the form of ax + b = 0 is called simple linear equation which gives a unique solution. It has only one variable x with power +1 and the variable is associated with other constant a and b.

   2. Linear Equation

   The equation in which all the variables have the power +1 is a linear equation. There may be one or more variables associated with constants. For example, ax + by + c = 0 is a linear equation, because the powers of variables x and y is +1.

   3. Quadratic Equation

   The equation in the form of ax² + bx + c = 0 is said to be quadratic equation. To be quadratic equation the highest power of the variable is +2.

   4. Polynomial Equation

   The equation with a higher power of the variables is called polynomial equation. a₁xⁿ + a₂x^n-1 + ..... a_n = 0 is a polynomial equation. Here the given equation is the n^th degree polynomial equation because the highest power of x is n.

   5. Simultaneous Equation

   The equation system which has at least two variables associated with each other is called simultaneous equation. For example,

   x + y = 0 ............ (i)

   2x + y = 0 ............ (ii)

   The equation (i) and (ii) are two variable simultaneous equation. They have only two variables x and y.

   Note: The number of a simultanious equation in the system should be equal to the number of variables.

   Solution of Simultaneous equations

   There are many methods to solve the simultaneous equation.But here we study only some important and useful method as given below:

   (a) Substitution Method
   (b) Elimination Method or Addition and Subtraction Method
   (c) Cross Multiplication Method

   But here westudy only two method.

   (a) Substitution Method

   In this method, the value of one variable is calculated in terms of another and the value is substituted in next equation.

   For example:
   x + y = 1
   or, y = 1 - x
   or, x = 1 - y

   Substitute the value of x or y in other equation and solve it.

   (b) Elimination Method or Addition and Subtraction Method

   In this method, one of the variables from the given equations is eliminated by adding or subtracting the given equations. Then the two equation are reduced to one equation with only one variable to find out the value of the variable.

   For example:

   2x + 4y -6= 0 ................... (i)

   10x + 10y +20 = 0 .............. (ii)

   Multiply both side of the equation (i) by 5, we get

   10x + 20y - 30 = 0 ............(iii)

   Subtracting equation (ii) from (iii), we get

    

   	10x	+	20y	-	30	=	0
   	10x	+	10y	+	20	=	0
   -		-		-
   			10y	-	50	=	0

   or, 10y - 50 = 0

   or, 10y =50

   or, y = \(\frac{50}{10}\)

   or, y =5

   Substituting value of y in equation (i) we get,

   2x + 8y - 6 = 0

   or, 2x + 8(5) - 6 = 0

   or, 2x + 40 = 6

   or, 2x = 6 - 40

   or, 2x = - 34

   or, x = \(\frac{-34}{2}\)

   or x = -17

   ∴x = -17 and y = 5

   Solution of Quadratic Equation

   Let ax + b = 0 be an equation in which, a≠ 0, x is a variable with index 1, a and b are constants.

   When this equation is solved, we get x = - \(\frac {b}{a}\)

   The equation ax + b = 0 is called linear equation.
   Let's take another example, ax² + bx + c = 0, where a≠ 0, in which x is a variable with 2 as its highest index and a, b, c are constants. The equation of this type is called quadratic equation.