Exponential And Logarithmic Functions Explained: Edexcel IGCSE Further Pure Mathematics Complete Breakdown

Exponential and logarithmic functions are foundational to the Edexcel IGCSE Further Pure Mathematics specification and appear consistently across both written papers. This article provides a complete treatment of both function types, the laws that govern them, and the techniques needed to solve equations involving them.

If you have encountered exponential growth in science or compound interest in economics, you have already met exponential functions in disguise. The Pearson Edexcel IGCSE Further Pure Mathematics course formalises these ideas and pairs them with their inverse: the logarithm. Together, exponential and logarithmic functions form a toolkit that connects to nearly every other section of the specification, from series to calculus. Understanding what is exponential and logarithmic functions igcse level content is essential for performing well in the exam.

This article serves as a comprehensive set of edexcel igcse further pure mathematics notes on the topic. Whether you are building your understanding from scratch or looking for IGCSE 4PM1 edexcel igcse exponential and logarithmic functions revision material, the content below covers the edexcel igcse further pure mathematics definition of each concept, the laws you must know, and worked examples that mirror the patterns found in the exam. The exponential and logarithmic functions explained here follow the specification precisely.

Key facts

The exponential function y = a^x

The exponential function with base a (where a is a natural number greater than 1) has several defining properties:

• It passes through the point (0, 1), because a⁰ = 1 for any a
• It is always positive: a^x > 0 for all real values of x
• It increases without bound as x increases
• It approaches zero as x decreases, but never reaches it (the x-axis is a horizontal asymptote)
• It is a one-to-one function, which is why its inverse (the logarithm) exists

The special case where a = e (Euler's number, approximately 2.718) produces the natural exponential function y = e^x, which has the unique property that its derivative equals itself. This makes it central to the calculus section of the specification.

The logarithmic function y = log_a(x)

The logarithm base a of x answers the question: "To what power must a be raised to produce x?" In symbols, if a^y = x, then y = log_a(x).

The graph of y = log_a(x) is the reflection of y = a^x in the line y = x. Its properties mirror those of the exponential:

• It passes through (1, 0), because log_a(1) = 0
• It is defined only for x > 0
• It increases slowly and without bound
• It has a vertical asymptote at x = 0 (the y-axis)
• log_a(a) = 1, because a¹ = a

The laws of logarithms

The specification requires mastery of four laws, plus the change of base formula. These are the tools that allow you to simplify, expand and solve logarithmic expressions.

Law 1: The product law

log_a(xy) = log_a(x) + log_a(y)

Law 2: The quotient law

log_a(x/y) = log_a(x) - log_a(y)

Law 3: The power law

log_a(x^k) = k log_a(x)

Law 4: Special values

log_a(a) = 1 and log_a(1) = 0

Change of base formula

log_a(x) = log_b(x) / log_b(a)

This formula is particularly useful when your calculator only has log base 10 or natural log (ln) buttons. To evaluate log₃(7), for instance, you compute log(7) / log(3) = 0.8451 / 0.4771 = 1.771 (to 4 significant figures).

Solving exponential equations

Equations of the form a^x = b are solved by taking logarithms of both sides.

Worked example 1

Solve 5^x = 20.

Take log base 10 of both sides:

log(5^x) = log(20)

x log(5) = log(20)

x = log(20) / log(5)

x = 1.3010 / 0.6990

x = 1.861 (to 4 significant figures)

Worked example 2

Solve 3^2x+1 = 45.

Take logarithms:

(2x + 1) log(3) = log(45)

2x + 1 = log(45) / log(3)

2x + 1 = 1.6532 / 0.4771

2x + 1 = 3.465

2x = 2.465

x = 1.232 (to 4 significant figures)

Solving logarithmic equations

Logarithmic equations are solved by combining logarithmic terms using the laws, then converting to exponential form.

Worked example 3

Solve log₂(x) + log₂(x + 6) = 4.

Step 1: Combine using the product law.

log₂(x(x + 6)) = 4

Step 2: Convert to exponential form.

x(x + 6) = 2⁴ = 16

Step 3: Expand and solve the quadratic.

x² + 6x - 16 = 0

(x + 8)(x - 2) = 0

x = -8 or x = 2

Step 4: Check validity. log₂(-8) is undefined, so x = -8 is rejected.

Answer: x = 2

Worked example 4

Solve log₅(2x - 1) - log₅(x + 3) = 1.

Step 1: Apply the quotient law.

log₅((2x - 1) / (x + 3)) = 1

Step 2: Convert to exponential form.

(2x - 1) / (x + 3) = 5¹ = 5

Step 3: Solve the resulting equation.

2x - 1 = 5(x + 3)

2x - 1 = 5x + 15

-3x = 16

x = -16/3

Step 4: Check validity. 2(-16/3) - 1 = -35/3, which is negative. log₅ of a negative number is undefined, so this equation has no solution.

Properties of indices (review)

The laws of logarithms derive from the laws of indices, so fluency with both is essential. The specification requires the following index laws:

Surds and rationalising the denominator

The specification also requires manipulation of surds (expressions involving square roots or other roots that cannot be simplified to rational numbers) and the technique of rationalising the denominator.

To rationalise a denominator of the form a + sqrt(b), multiply both numerator and denominator by the conjugate a - sqrt(b).

Worked example 5

Simplify 6 / (3 + sqrt(5)).

Multiply by the conjugate:

6(3 - sqrt(5)) / ((3 + sqrt(5))(3 - sqrt(5)))

= 6(3 - sqrt(5)) / (9 - 5)

= 6(3 - sqrt(5)) / 4

= 3(3 - sqrt(5)) / 2

= (9 - 3sqrt(5)) / 2

Worked example 6

Express sqrt(48) + sqrt(12) in the form a sqrt(3).

sqrt(48) = sqrt(16 times 3) = 4sqrt(3)

sqrt(12) = sqrt(4 times 3) = 2sqrt(3)

sqrt(48) + sqrt(12) = 4sqrt(3) + 2sqrt(3) = 6sqrt(3)

Exam question patterns

The edexcel igcse further pure mathematics explained material on exponential and logarithmic functions typically appears in the following question patterns:

1. Solve a^x = b: Take logs, apply the power law, isolate x.
2. Simplify a logarithmic expression: Apply the product, quotient and power laws to combine or expand terms.
3. Solve a logarithmic equation: Combine logs, convert to exponential form, solve the resulting equation, check validity.
4. Sketch graphs: Draw y = a^x or y = log_a(x), marking key features (intercepts, asymptotes).
5. Change of base: Convert between logarithm bases to evaluate or compare expressions.
6. Prove a logarithmic identity: Start from one side and use the laws to reach the other.

Self-check questions

Test your understanding with the following practice problems:

1. Solve 4^x = 30, giving your answer to 3 significant figures.
2. Simplify 2 log₃(5) + log₃(4) - log₃(100) into a single logarithm.
3. Solve log₂(3x - 1) = 5.
4. Rationalise the denominator of 10 / (sqrt(7) - sqrt(2)).
5. Express log₄(x) in terms of log₂(x) using the change of base formula.
6. Solve log₁₀(x²) + log₁₀(x) = 6.
7. Simplify sqrt(75) - 2sqrt(27) + sqrt(48).

Self-check questions

1. Solve 2^3x-1 = 16 without a calculator, showing each step.
2. Given log₃(x) + log₃(x - 2) = 1, find the value of x.

Mastery of edexcel igcse exponential and logarithmic functions requires consistent practice with the laws of logarithms and the technique of converting between exponential and logarithmic forms. The patterns are predictable, the laws are few, and the errors students make are well documented. Work through the examples above, attempt the self-check questions, and then use the Green Bridge CBT platform for additional practice questions organised by topic. The more familiar you become with these function types, the more confident you will feel when they appear in the exam.