Probability

INDICES
We recall that a power is the product of a certain number of factors, all of which are the same. For example, 37 is a power, in which the number 3 is called the base and the number 7 is called the index or exponent.

Laws of Indices

1

To multiply powers with the same base, add the indices.
aman = am+n. 

2

To divide powers with the same base, subtract the indices.
 = am − n, (provided m > n.)

3

To raise a power to a power, multiply the indices.
(am)n = amn.

4

A power of a product is the product of the powers.
(ab)m = ambm.

5

A power of a quotient is the quotient of the powers.
m= , (provided b ≠ 0.)

The Zero Index

Clearly  = 1. On the other hand, applying index law 2, ignoring the condition m > n,
we have  = 50. If the index laws are to be applied in this situation, then we need to define 50 to be 1.

More generally, if a ≠ 0 then we define a0 = 1.

Note that 00 is not defined. It is sometimes called an indeterminant form.

Negative Exponents

If we examine the pattern formed when we take decreasing powers of 2, we see

24 = 16, 23 = 8, 22 = 2, 21 = 2, 20 = 1, 2−1 = ?, 2−2 = ?

At each step as we decrease the index, the number is halved. Thus it is sensible to define

2−1 = .

Furthermore, continuing the pattern, we define

2−2 =  = , 2−3 =  = , and so on.

These definitions are consistent with the index laws.

Fractional Indices

We now extend our study of indices to include rational or fractional exponents. In particular, can we give meaning to 4?

Once again, we would like the established index laws to hold. Hence, squaring this expression we would like to say:

42 = 4 × 2 = 41 = 4.

Thus we define 4 to be  = 2.

In general we define a =  for any positive number a.

Note that we have defined a to be the positive square root of We do this so that there is only one value for a.

Negative fractional indices

Finally, we can extend the indices to include negative rationals. For example,

8− = 8−1 =  = .

So that

a = .

LOGARITHMS

Logarithms appear in all sorts of calculations in engineering and science, business and economics.

Before the days of calculators they were used to assist in the process of multiplication by replacing

the operation of multiplication by addition. Similarly, they enabled the operation of division to

be replaced by subtraction. They remain important in other ways, one of which is that they

provide the underlying theory of the logarithm function. This has applications in many fields, for

example, the decibel scale in acoustics.

Consider the expression 16 = 2^4

. Remember that 2 is the base, and 4 is the power. An alternative,

yet equivalent, way of writing this expression is log2 16 = 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 = 2^4

log2 16 = 4

The relationship connecting logarithms and powers is:

x = loga y means y = ax.

The number is called the base and must be a positive number. Also since ax is positive, we can only find the logarithm of a positive number. We will assume from now on that both are positive, but can be negative.

Laws of Logarithms

1  loga = 0 and loga a = 1

2  If x and y are positive numbers, then loga xy = loga x + loga y

    That is, the logaithm of a product is the sum of the logarithms.

           Suppose x = ac and y = ad so that loga x = c and loga y = d.

Then		xy	= ac × ad
			=ac+d			(by Index law 1)
So		loga xy		= loga ac+d
				= c + d
				= loga x + loga y

3 If x and y are positive numbers, then loga  = loga x − loga y.

That is, the logarithm of a quatient is the difference of their logarithms.

Suppose x = ac and y = ad so that loga x = c and loga y = d.

Then			=
			= ac−d			(by Index law 2)
So		loga		= loga ac−d
				= c − d
4				= loga x + loga y// If x is a positive number, then loga  = −loga x.

       This follows from logarithm law 3 and logarithm law 1.

	loga	= loga 1 − loga x		(logarithm law 3)
		= 0 − loga x		(logarithm law 1)
		= −loga x, as required.

5  If x is a positive number and n is any rational number, then loga (xn) = nlogax.

    This follows from logarithm law 3 and logarithm law 1.

	loga (xn)	= loga ((ac)n)
		= loga (acn)		(by Index law 3)
		= nloga x, as required. Change of base Some calculators are able to find the logarithm of a number to any positive base. This is not, however, universal, and there are many occasions when we would like to change from one base to another. The change of base formula states that: logb c = .