Logarithmic functions and indices for Edexcel IGCSE Further Pure Mathematics: Methods and Worked Examples

Logarithms and indices are the gateway topic for Edexcel IGCSE Further Pure Mathematics, and they feed into almost everything that follows

Think of logarithms as a translator. You already know how to speak in powers: 2³ = 8. A logarithm simply asks the question in reverse: "What power of 2 gives 8?" The answer is log₂ 8 = 3. That reverse question turns up everywhere in this edexcel igcse further pure mathematics course, from solving exponential equations to sketching curves, and from simplifying expressions to working with series.

This article is your set of edexcel igcse further pure mathematics logarithmic functions and indices revision notes, covering the entire section. It covers exponential and logarithmic functions, the properties of indices and logarithms, surds, and rationalising the denominator. Each topic includes worked examples and common pitfalls so you can build confidence before the exam.

Exponential and logarithmic functions

The specification requires you to know the functions a^x and log_b x where b is a natural number greater than 1, including the shape of their graphs.

The graph of y = a^x (where a > 1) passes through (0, 1) because any positive number raised to the power 0 equals 1. It rises steeply for positive x and approaches the x-axis (but never touches it) as x becomes large and negative. The x-axis is a horizontal asymptote.

The graph of y = log_b x is the reflection of y = b^x in the line y = x. It passes through (1, 0) because log_b 1 = 0, and through (b, 1) because log_b b = 1. It is defined only for x > 0, and the y-axis is a vertical asymptote.

Properties of indices and logarithms

The laws of logarithms are examined consistently in edexcel igcse further pure mathematics. You must know them fluently and be able to apply them in both directions: combining separate logarithms into a single one, or expanding a single logarithm into several.

Worked example: combining logarithms

Simplify 2 log₃ 4 - log₃ 2 + log₃ 5.

Apply the power law first: 2 log₃ 4 = log₃(4²) = log₃ 16.

Combine using the product and quotient laws: log₃ 16 - log₃ 2 + log₃ 5 = log₃(16/2) + log₃ 5 = log₃ 8 + log₃ 5 = log₃(8 x 5) = log₃ 40.

Solving equations of the form a^x = b

When the variable is in the exponent, take logarithms of both sides. The base you choose does not matter, but log₁₀ or ln are most convenient because calculators have those keys.

Worked example: Solve 5^x = 20.

Take log₁₀ of both sides: x log₁₀ 5 = log₁₀ 20.

So x = log₁₀ 20 / log₁₀ 5 = 1.3010... / 0.6990... = 1.861 (3 s.f.).

Change of base formula

The change of base formula states that log_a x = log_b x / log_b a. This is essential when your calculator only has log₁₀ and ln keys, or when you need to compare logarithms in different bases.

Worked example: Evaluate log₈ 32.

Using change of base to base 2: log₈ 32 = log₂ 32 / log₂ 8 = 5/3.

Surds

A surd is an irrational root left in exact form rather than rounded to a decimal. Keeping answers as surds is like keeping fractions rather than decimals: you preserve exactness. The edexcel igcse further pure mathematics exam frequently asks for answers in surd form, so fluency with surd manipulation is a practical necessity.

The key rules are:

• \u221a(ab) = \u221aa x \u221ab
• \u221a(a/b) = \u221aa / \u221ab
• (\u221aa)² = a

Worked example: simplifying surds

Simplify \u221a72 + 3\u221a8 - \u221a18.

Break each surd into a product involving a perfect square:

\u221a72 = \u221a(36 x 2) = 6\u221a2

3\u221a8 = 3\u221a(4 x 2) = 3 x 2\u221a2 = 6\u221a2

\u221a18 = \u221a(9 x 2) = 3\u221a2

Combining: 6\u221a2 + 6\u221a2 - 3\u221a2 = 9\u221a2.

Rationalising the denominator

Rationalising means removing the surd from the denominator of a fraction. There are two standard techniques, depending on whether the denominator is a single surd or a sum/difference involving a surd.

Case 1: single surd denominator

Multiply numerator and denominator by the surd.

Worked example: Rationalise 5 / \u221a3.

Multiply top and bottom by \u221a3: (5 x \u221a3) / (\u221a3 x \u221a3) = 5\u221a3 / 3.

Case 2: compound denominator

Multiply by the conjugate. The conjugate of (a + \u221ab) is (a - \u221ab).

Worked example: Rationalise 4 / (3 + \u221a5).

Multiply top and bottom by (3 - \u221a5):

Numerator: 4(3 - \u221a5) = 12 - 4\u221a5.

Denominator: (3 + \u221a5)(3 - \u221a5) = 9 - 5 = 4.

Result: (12 - 4\u221a5) / 4 = 3 - \u221a5.

Practice questions to test yourself

These edexcel igcse further pure mathematics practice questions cover the main techniques from this section. Work through them without looking at the solutions above, then check your method.

1. Solve the equation 3^2x-1 = 15, giving your answer to 3 significant figures.
2. Simplify log₄ 8 + log₄ 2 - log₄ 32, leaving your answer as a single logarithm or an integer.
3. Show that (2 + \u221a3)(2 - \u221a3) = 1, and hence rationalise 1 / (2 - \u221a3).
4. Given that log_a 3 = p and log_a 5 = q, express log_a 45 in terms of p and q.
5. Simplify \u221a50 - 2\u221a8 + 3\u221a32, giving your answer in the form k\u221a2.

Solutions

Question 1: 3^2x-1 = 15. Take log₁₀ of both sides: (2x - 1) log₁₀ 3 = log₁₀ 15. So 2x - 1 = log₁₀ 15 / log₁₀ 3 = 1.17609... / 0.47712... = 2.4649... Therefore 2x = 3.4649..., and x = 1.73 (3 s.f.).

Question 2: log₄ 8 + log₄ 2 - log₄ 32 = log₄(8 x 2 / 32) = log₄(16/32) = log₄(1/2). Using change of base: log₄(1/2) = log₂(1/2) / log₂ 4 = (-1)/2 = -1/2.

Question 3: (2 + \u221a3)(2 - \u221a3) = 4 - 3 = 1. Therefore 1 / (2 - \u221a3) = (2 + \u221a3) / [(2 - \u221a3)(2 + \u221a3)] = (2 + \u221a3) / 1 = 2 + \u221a3.

Question 4: 45 = 9 x 5 = 3² x 5. So log_a 45 = log_a(3²) + log_a 5 = 2 log_a 3 + log_a 5 = 2p + q.

Question 5: \u221a50 = 5\u221a2, 2\u221a8 = 2 x 2\u221a2 = 4\u221a2, 3\u221a32 = 3 x 4\u221a2 = 12\u221a2. So 5\u221a2 - 4\u221a2 + 12\u221a2 = 13\u221a2.

Logarithmic functions and indices edexcel igcse form the algebraic backbone of this course. Once these edexcel igcse further pure mathematics notes are secure in your working memory, the topics that build on them (series, calculus, curve sketching) become substantially more accessible. Spend the time here, and the rest of the igcse 4PM1 logarithmic functions and indices material will follow naturally.

Self-check questions

1. Simplify log₂(32) without a calculator.
2. Solve 3^x = 81 by converting to logarithmic form.
3. Given that log_a(x) = 2 and log_a(y) = 5, find log_a(x²y).
4. Sketch the graph of y = 2^x for values of x from -3 to 3. Mark the y-intercept and describe the behaviour as x approaches negative infinity.
5. The population of a colony of bacteria doubles every 4 hours. If the initial population is 500, write an expression for the population after t hours and find how long it takes to reach 16,000.

For further edexcel igcse further pure mathematics explained content on each sub-topic, and for more practice across all ten content areas, use edexcel igcse further pure mathematics revision notes on the Green Bridge CBT platform to track your progress and focus your revision where it matters most.