Mathematics Form 1-4 : CHAPTER TWENTY SIX – INDICES AND LOGARITHMS

Specific Objectives

By the end of the topic the learner should be able to:

1. Define indices (powers)
2. State the laws of indices
3. Apply the laws of indices in calculations
4. Relate the powers of 10 to common logarithms
5. Use the tables of common logarithms and anti-logarithms in computation.

Content

1. Indices (powers) and base
2. Laws of indices (including positive integers, negative integers and fractional indices)
3. Powers of 10 and common logarithms
4. Common logarithms:

• characteristics
• mantissa

1. Logarithm tables
2. Application of common logarithms in multiplication, division, powers and roots.

Introduction

Index and base form

The power to which a number is raised is called index or indices in plural.

=

5 is called the power or index while 2 two is the base.

100 =

2 is called the index and 10 is the base.

Laws of indices

For the laws to hold the base must be the same.

Rule 1

Any number, except zero whose index is 0 is always equal to 1

Example

=1

Rule 2

To multiply an expression with the same base, copy the base and add the indices.

Example

=

= 3125

Rule 3

To divide an expression with the same base, copy the base and subtract the powers.

Example

Rule 4

To raise an expression to the nth index, copy the base and multiply the indices

Example

) ²

=

Rule 5

When dealing with a negative power, you simply change the power to positive by changing it into a fraction with 1 s the numerator.

=

Example

=

Example

Evaluate:

=

=1

1. (() 2

=()

=1

=1 2 or =) squared =

Fractional indices

Fractional indices are written in fraction form. In summary if. a is called the root of b written as .

Example

= = () = = 8

=3

=

=

LOGARITHM

Logarithm is the power to which a fixed number (the base) must be raised to produce a given number. = n is written as =m.

= n is the index notation while = m is the logarithm notation.

Examples

Reading logarithms from the tables is the same as reading squares square roots and reciprocals.

We can read logarithms of numbers between 1 and 10 directly from the table. For numbers greater than 10 we proceed as follows:

Express the number in standard form, A X .Then n will be the whole number part of the logarithms.

Read the logarithm of A from the tables, which gives the decimal part of the logarithm. Then add it to n which is the power of 10 to form the positive part of the logarithm.

Example

Find the logarithm of:

379

Solution

379

= 3.79 x

Log 3.79 = 0.5786

Therefore the logarithm of 379 is 2 + 0.5786= 2.5786

The whole number part of the logarithm is called the characteristic and the decimal part is the mantissa.

Logarithms of Positive Numbers less than 1

Example

Log to base 10 of 0.034

We proceed as follows:

Express 0.034 in standard form, i.e., A X.

Read the logarithm of A and add to n

Thus 0.034 = 3.4 x

Log 3.4 from the tables is 0.5315

Hence 3.4 x =

Using laws of indices add 0.5315 + -2 which is written as.

It reads bar two point five three one five. The negative sign is written directly above two to show that it’s only the characteristic is negative.

Example

Find the logarithm of:

0.00063

Solution

(Find the logarithm of 6.3)

.7993

ANTILOGARITHMS

Finding antilogarithm is the reverse of finding the logarithms of a number. For example the logarithm of 1000 to base 10 is 3. So the antilogarithm of 3 is 1000.In algebraic notation, if

Log x = y then antilog of y = x.

Example

Find the antilogarithm of .3031

Solution

Let the number be x

X

(Find the antilog, press shift and log then key in the number)

Example

Use logarithm tables to evaluate:

Number Standard form logarithm

456 4.56 x 2.6590

398 3.98 x 2.5999

5.2589

271 2.71 x 2.4330

2.8259

= 669.7

To find the exact number find the antilog of 2.8259 by letting the characteristic part to be the power of ten then finding the antilog of 0.8259

Example

Operations involving bar

Evaluate

Solution

Example

= (9.45 x

= ( )

Note;

In order to divide .9754 by 2 , we write the logarithm in search away that the characteristic is exactly divisible by 2 .If we are looking for the root , we arrange the characteristic to be exactly divisible by n)

.9754 = -1 + 0.9754

= -2 + 1.9754

Therefore, .9754) =

= -1 + 0.9877

= .9877

Find the antilog of by writing the mantissa as power of 10 and then find the antilog of characteristic.

= 0.9720

Example

Number logarithm

+ 1.7910)

3.954 x . 5970 (find the antilog)

0.3954

End of topic  

Past KCSE Questions on the cubes, cubes roots, Reciprocals indices and logarithms.

1. Use logarithms to evaluate

3 36.15 x 0.02573

1,938

1. Find the value of x which satisfies the equation.

16^x2 = 8^4x-3  

1. Use logarithms to evaluate ( 1934)² x √ 0.00324

   436

2. Use logarithms to evaluate

   55.9 ÷ (02621 x 0.01177) ^1/5

3. Simplify  2^x x 5^2x¸ 2^-x
4. Use logarithms to evaluate

   (3.256 x 0.0536)^1/3

    

5. Solve for x in the equation  

32^(x-3) ÷8 ^(x-4) = 64 ÷2^x

1. Solve for x in the equations 81^2x x 27^x = 729

   9x

2. Use reciprocal and square tables to evaluate to 4 significant figures, the expression:

   1 + 4 .346²

24.56

1. Use logarithm tables, to evaluate

   

   0.032 x 14.26 ^2/3

   0.006

    

2. Find the value of x in the following equation

   49^{(x +1)} + 7^(2x) = 350

    

3. Use logarithms to evaluate

   (0.07284)²

   3√0.06195

4. Find the value of m in the following equation

   (1/27^m x (81)^-1 = 243

5. Given that P = 3^y express the equation 3^(2y-1) + 2 x 3 ^(y-1) = 1 in terms of P hence or otherwise find the value of y in the equation 3 ^{(2y – 1)} + 2 x 3 ^(y-1) = 1

1. Use logarithms to evaluate 55.9 ¸(0.2621 x 0.01177)^1/5
2. Use logarithms to evaluate

6.79 x 0.3911^¾

 Log 5

1. Use logarithms to evaluate

3   1.23 x 0.0089

79.54

1. Solve for x in the equation

   X = 0.0056 ^½

    1.38 x 27.42