Mathematics · Lesson 3

Indices & logs without fear

JAMBWAECNECO

What the syllabus asks

JAMB: Number and numeration: indices, logarithms and surds

WAEC: Indices (laws of indices, zero and negative indices, fractional indices, simple indicial equations) and Logarithms (laws of logarithms, change of base, logarithmic equations)

NECO: Indices (laws of indices, zero and negative indices, fractional indices, simple indicial equations) and Logarithms (laws of logarithms, change of base, logarithmic equations)

What you will learn

State and use the laws of indices.
Evaluate zero, negative and fractional indices without a calculator.
Solve indicial equations by equating bases.
Convert between index form and logarithm form.
Use the laws of logarithms to simplify and evaluate expressions.
Solve logarithmic equations and use change of base.

Key points

aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ.
a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(m/n) = (ⁿ√a)ᵐ, root first.
Indicial equations: write both sides with the same base, then equate the indices.
logₐN = x ⇔ aˣ = N; logₐa = 1 and logₐ1 = 0.
log xy = log x + log y; log(x/y) = log x − log y; log xⁿ = n log x.
Check log-equation answers: logs of zero or negative numbers are undefined.
Change of base: logₐN = log N/log a.

Start here: What an index really means

Imagine a pyramid scheme (a bad idea, by the way). You tell 2 friends. Each of them tells 2 friends, and each of those tells 2 more. After 5 rounds, how many people are in the newest round? It is 2 × 2 × 2 × 2 × 2 = 32. Writing "2 × 2 × 2 × 2 × 2" is slow, so mathematicians write 2⁵.

Definition. aⁿ = a × a × a × … × a (n factors). The number a is the base and n is the index (or power).

So 3⁴ = 3 × 3 × 3 × 3 = 81, and 10³ = 1000. Note that 3⁴ is NOT 3 × 4. The index counts how many factors there are; it is not something you multiply by.

Figure 3.1: The index counts the factors.

Worked example 3.1

Evaluate (a) 5³ (b) (−2)⁴ (c) −2⁴.

(a) 5 × 5 × 5 = 125.
(b) (−2)(−2)(−2)(−2) = 16. An even number of negatives multiplies to a positive.
(c) −2⁴ means −(2⁴) = −16. Without brackets, the index belongs to the 2 only, not to the minus sign.

125; 16; −16

⚠️ Common mistake: treating (−2)⁴ and −2⁴ as the same. Brackets decide what the index applies to.

Try it yourself. Write 2 × 2 × 2 × 3 × 3 in index form and evaluate it.

Show solution

2³ × 3² = 8 × 9 = 72.

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Practice questions

1. Evaluate 125^(−2/3).

Show worked answer

³√125 = 5, 5² = 25, so 1/25.

2. Simplify (2a³b)² × 3a⁻¹b.

Show worked answer

4a⁶b² × 3a⁻¹b = 12a⁵b³.

3. Solve 27ˣ = 9ˣ⁺¹.

Show worked answer

3x = 2x + 2, so x = 2.

4. Given log 2 = 0.3010 and log 3 = 0.4771, find log 7.2.

Show worked answer

7.2 = 2³ × 3² ÷ 10, so 0.9030 + 0.9542 − 1 = 0.8572.

5. Solve log₂x + log₂(x + 2) = 3.

Show worked answer

x² + 2x − 8 = 0, so x = 2 or −4. Reject −4: x = 2.

Key words

Base: The number being multiplied repeatedly, such as 3 in 3⁴.
Index (power, exponent): The small raised number showing how many times the base is used as a factor.
Indices: The plural of index.
Reciprocal: 1 divided by a number; the reciprocal of 5 is 1/5.
Root: A number which, raised to a power, gives the original; ³√8 = 2 because 2³ = 8.
Logarithm: The power to which a base must be raised to give a number.
Common logarithm: A logarithm to base 10, written log or lg.
Indicial equation: An equation in which the unknown is in the index.
Antilogarithm: The number whose logarithm is given; the antilog of 2 (base 10) is 100.
Change of base: Rewriting a log in a new base: logₐN = log N ÷ log a.

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