1. (a) Identify and outline the sub-sub-strands: Indices, Logarithms to base 10, Application of indices and logarithms.
2. (b) Express numbers in index form.
3. (c) Derive the laws of indices using factors.
4. (d) Apply the laws of indices in different situations.
5. (e) Relate index notation to logarithm notation to base 10.
6. (f) Determine common logarithms of numbers from mathematical tables and calculators.
7. (g) Apply common logarithms in multiplication, division, powers and roots of numbers.
8. (h) Appreciate the use of indices and common logarithms in mathematical computations.

1. Sub-sub-strands

• Indices — index form, integer and fractional indices, zero and negative indices, laws of indices.
• Logarithms (base 10) — definition of common logarithm, characteristic and mantissa, use of tables and calculators.
• Applications — using indices and logs to simplify multiplication, division, powers and roots, and real-life calculations (e.g., scientific notation).

2. Index (power) notation — basic ideas

To write repeated multiplication in compact form we use indices (exponents). Example:

General form: aⁿ means a multiplied by itself n times (a is the base, n is the index or exponent).

Express numbers in index form

• 36 = 6 × 6 = 6²
• 125 = 5 × 5 × 5 = 5³
• 1000 = 10 × 10 × 10 = 10³ (useful for scientific notation)

3. Derive the laws of indices using factors

We show common laws by writing each power as repeated factors.

4. Apply the laws of indices — examples

1. Simplify 3⁵ ÷ 3² = 3³ = 27.
2. Simplify (x² y³)² = x⁴ y⁶.
3. Simplify (8⁻¹) × 8³ = 8⁻¹⁺³ = 8² = 64.
4. Express 0.00045 in index (scientific) form: 4.5 × 10⁻⁴.

5. Logarithms (base 10) — definition and relation to indices

If 10^x = N, then the common logarithm (logarithm to base 10) of N is x: log₁₀N = x. We usually write log N for log₁₀N.

So indices and logarithms are inverse ideas:

Characteristic and mantissa

For common logs of positive numbers:

• Characteristic = integer part of log (can be negative or zero).
• Mantissa = fractional part (always non-negative for positive N).

Example: log 23 ≈ 1.3617 → characteristic 1, mantissa 0.3617.

6. Using mathematical tables and calculators

Using a table (common log table)

1. Write number in standard form so first digit is non-zero: e.g., 23 = 2.3 × 10¹. Then log 23 = 1 + log 2.3.
2. Look up mantissa for 2.3 in the log table (approx 0.3617), so log 23 = 1.3617.
3. Tables usually give mantissa for the first few digits; you may interpolate for more accuracy.

Using a calculator

Use the key labeled "log". Example: press log then 23 → display ≈ 1.361727836.

7. Apply common logarithms to computations

Useful identities:

• log(ab) = log a + log b → multiply numbers by adding logs.
• log(a/b) = log a − log b → divide numbers by subtracting logs.
• log(aⁿ) = n log a → powers by multiplication.
• log(√[n]{a}) = (1/n) log a → roots by division.

8. Classroom activities and suggested learning experiences (Kenyan context)

• Group task: Give groups several large multiplications (e.g., 234 × 987) — students use log tables and calculators to check results; compare methods and accuracy.
• Hands-on derivation: Students write out factors for small exponents (2⁴, 2⁶) to see why laws of indices work, then present proofs on the board.
• Practical application: Convert measurements to scientific notation (e.g., distance in metres, mass in kg) and use indices to compare very large/small numbers.
• Table practice: Teach how to find mantissa and characteristic in a standard log table — practice with interpolation for 3-significant-digit values.
• Calculator skills: Practice using the log key, 10^x (antilog) keys, and understand rounding and significant figures.
• Exam-style questions: Provide past KCSE-like questions on indices and logs for timed practice.
• Discussion/reflection: Ask students why logs used to be essential before cheap calculators and where we still use logarithms today (e.g., Richter scale, pH, sound intensity, computer science algorithms, scientific notation).

9. Exercises (with short answers)

1. Express 0.00032 in index form. Answer: 3.2 × 10⁻⁴.
2. Simplify: x⁵ × x⁻². Answer: x³.
3. Simplify: (2a² b)³. Answer: 8 a⁶ b³.
4. Find log 250 using log tables / calculator (show steps). Answer: log 250 = log(2.5×10²) = 2 + log 2.5 ≈ 2 + 0.39794 = 2.39794.
5. Use logs to compute 64 × 125. (Hint: use log 64 = 1.80618, log 125 = 2.09691.) Answer: sum = 3.90309 → antilog 0.90309 ≈ 8.00 × 10^3−? → direct product = 8000 (since 64×125 = 8000).
6. Compute √[3]{216} using indices. Answer: 216 = 6³ so cube root = 6.

10. Appreciation and summary

Indices give a compact way to handle repeated multiplication and very large or very small numbers. Common logarithms (base 10) turn multiplication and powers into addition and multiplication, which historically made calculations much simpler using tables. Today, calculators provide quick logs and antilogs, but understanding the relationship between indices and logs helps in estimation, scientific notation and many real-world scales (e.g., sound, earthquakes, acidity).

Good practice: learn the index laws by factor reasoning, practise using log tables at least once, and become fluent with your calculator's log and 10^x keys.