Indices and Logarithms

Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 1.0 Numbers | Sub-Strand 1.3 Indices and Logarithms (6 Lessons)

Lesson objectives

By the end of this sub-strand, you should be able to:

• State and apply the laws of indices in different situations.
• Interpret zero, negative and fractional indices correctly.
• Relate indices to logarithms and move between the two forms.
• Use logarithms to simplify calculations involving multiplication, division and powers.

Indices and Logarithms

Writing 2 × 2 × 2 × 2 × 2 × 2 × 2 gets tiring. Writing 2⁷ says the same thing in two characters.

Indices are a shorthand for repeated multiplication. Logarithms are the same idea read backwards.

a) Index form

The base is the number being multiplied. The index, also called the power, tells you how many times.

Say 2⁴ as "two to the power four". It equals 16, not 8. A common slip is multiplying the base by the index instead of repeating the multiplication.

b) The laws of indices

Three laws cover almost every question. All three need the same base on both sides.

The rule that catches people is the middle one. Division subtracts, it does not divide. 3⁵ ÷ 3² is 3³, never 3².

If the bases differ, none of the laws apply. 2³ × 5³ cannot be simplified by adding indices, because 2 and 5 are different bases.

c) Two cases worth learning

Anything to the power zero equals 1. That includes 100⁰ and 7⁰, both of which are 1.

A negative index does not make the answer negative. It turns the number upside down. 2⁻³ is one eighth, a small positive number.

d) Powers of ten

Our number system is built on tens, so powers of 10 deserve their own look.

Notice the pattern. The index tells you how many zeros follow the 1.

e) Common logarithms

A logarithm asks the reverse question. Instead of "what is 10³", it asks "10 to what power gives 1000".

So log 1000 is 3, because 1000 is 10³. Not every number is an exact power of 10. Then the logarithm falls between two whole numbers, and you read it from tables.

Every number between 10 and 100 has a logarithm between 1 and 2. That check will tell you quickly whether an answer is sensible.

f) Where this is used

Scientists write very large and very small numbers using powers of 10, which keeps them readable. Engineers measure sound in decibels, a logarithmic scale. Compound interest on a SACCO loan grows by repeated multiplication, which is an index.

Words to know

• Base -- the number being repeatedly multiplied in an index expression.
• Index (exponent, power) -- the number recording how many times the base is used as a factor.
• Reciprocal -- one divided by a number; a negative index produces one.
• Logarithm -- the index to which a stated base must be raised to give a particular number.
• Common logarithm -- a logarithm taken to base 10.

:::checkpoint Check yourself

1. Write 5 × 5 × 5 in index form.
2. Simplify 4⁶ × 4³.
3. What is 9⁰?
4. What is log 10000? :::

Bridge to practice

The exercises move from applying single index laws to interpreting zero, negative and fractional indices, and finally to converting between forms and combining logarithm laws. When a question mixes several laws, apply them one at a time and write the intermediate index rather than trying to reach the final value in a single step.