Indices and Logarithms
Numbers · Indices and Logarithms
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 3 Indices and Logarithms
Lesson objectives
By the end of this topic, you should be able to:
- State and apply the laws of indices in calculations.
- Interpret zero, negative and fractional indices.
- Relate the powers of 10 to common logarithms.
- Use logarithm tables to carry out multiplication and division.
Indices and Logarithms
An index tells you how many times a base is multiplied by itself. A logarithm works backwards: it asks what the index was.
a) The laws of indices
Multiplying the same base adds the indices, because you are simply counting all the factors together.
Dividing subtracts them, since the shared factors cancel.
A power of a power multiplies them.
These laws only apply when the bases are the same. You cannot combine 2³ and 3² this way.
b) Why a zero index gives 1
Anything divided by itself is 1.
By the division law, aⁿ ÷ aⁿ is aⁿ⁻ⁿ, which is a⁰.
Both routes describe the same quantity, so a⁰ must equal 1.
The rule is not arbitrary; it is what keeps the laws consistent.
c) Negative indices
A negative index means a reciprocal.
By the same reasoning, a² ÷ a⁵ is a⁻³. But cancelling directly gives 1/a³.
So a⁻³ = 1/a³.
A negative index does not make the number negative. 2⁻³ is 1/8, which is positive.
d) Fractional indices
A fractional index means a root.
Since (a^½)² = a¹ = a, the quantity a^½ must be the square root of a.
Likewise a^⅓ is the cube root, and a^(m/n) is the nth root of aᵐ.
e) Logarithms
A logarithm is an index. Writing 100 as 10² means log 100 = 2.
Common logarithms use base 10, which is why they suit our number system.
Because logs are indices, the index laws become log laws:
log(AB) = log A + log B.
log(A ÷ B) = log A − log B.
log(Aⁿ) = n log A.
f) Using logarithm tables
Write the number in standard form. The power of ten gives the characteristic, and the tables give the mantissa.
Take 3 460 = 3.46 × 10³. The characteristic is 3 and the mantissa comes from the table, so the log is 3.5391.
To multiply, add the logs and take the antilogarithm. To divide, subtract them.
This turns multiplication into addition, which is exactly why logarithms were invented.
For numbers below 1 the characteristic is negative, written with a bar over it. Only the characteristic is negative; the mantissa stays positive.
g) Where this is used
Compound interest and population growth. Scales such as pH and the Richter scale. Any formula where a quantity multiplies repeatedly.
Words to know
- Base -- the number being raised to a power.
- Index (power) -- the number showing how many times the base is multiplied by itself.
- Logarithm -- the power to which a base must be raised to give a number.
- Characteristic -- the whole number part of a common logarithm.
- Mantissa -- the decimal part of a common logarithm.
:::checkpoint Check yourself
- Simplify 3⁴ × 3².
- Simplify (2³)⁴.
- Write 5⁻² as a fraction.
- Explain why any non-zero number to the power zero equals 1. :::
Bridge to practice
The exercises begin with the three laws and matching bases, move through zero, negative and fractional indices, and finish with logarithms, tables and index equations. For every negative index, write the reciprocal explicitly before evaluating — that one habit prevents the commonest error in the topic.