How do I solve an equation like 2^x = 10?

Take logarithms of both sides; any base works, but base 10 or natural log is typical: log(2^x) = log(10). Apply the power rule to get x log(2) = log(10), then divide: x = log(10)/log(2). log(10) = 1, log(2) ≈ 0.3010, so x ≈ 3.322. You can verify by calculating 2^3.322 ≈ 10. Using natural logs gives the same result because ln(10)/ln(2) is identical by the change-of-base formula.

What is the difference between log and ln?

In A-Level contexts, ‘log’ typically means log base 10 (common logarithm), while ‘ln’ is the natural logarithm with base e ≈ 2.718. ln x is the inverse of e^x, so ln(e) = 1. Natural logs are preferred in calculus because the derivative of ln x is 1/x and the derivative of e^x is e^x. You can use either to solve exponential equations, but ln often simplifies expressions involving e.

Why is the derivative of ln x equal to 1/x?

Let y = ln x, which means x = e^y. Differentiate both sides with respect to x: the left gives 1, and the right gives e^y * dy/dx by the chain rule. So 1 = e^y * dy/dx, but e^y = x, so 1 = x * dy/dx, hence dy/dx = 1/x. This is a fundamental result that you'll use when integrating 1/x to get ln|x| + c.

How do I rearrange y = ab^x to get a straight-line graph?

Take logs of both sides: ln y = ln(ab^x) = ln a + ln(b^x) = ln a + x ln b. This is of the form Y = mX + c, with Y = ln y, m = ln b, X = x, and c = ln a. So plotting ln y against x gives a straight line. The gradient is ln b and the y-intercept is ln a. Once you find these from the graph or data, exponentiate: a = e^(c) and b = e^(m).

When should I use log laws in exam questions?

Use them whenever you see a combination of logs being added or subtracted, when simplifying logarithmic expressions, or when you need to take the log of both sides of an equation with an exponent. They are also essential for converting exponential models into linear form for modelling questions. In calculus, you might need to split a log expression using log laws before differentiating or integrating.

F: Exponentials and logarithms

AQA

A-Level

This topic covers the properties and graphs of exponential functions, including base 'a' and the natural exponential 'e'. It also introduces logarithms as the inverse of exponential functions, focusing on the laws of logarithms and solving exponential equations.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Topic Overview

The topic of exponentials and logarithms introduces you to the powerful inverse relationship between exponential growth and logarithmic scaling. You’ll learn to manipulate expressions involving powers and logs, solve equations where the unknown is in the exponent, and apply these skills to real-world models from biology, finance, and physics. This section fundamentally extends your GCSE work on indices and prepares you for advanced calculus involving e and ln x.

Understanding exponentials and logs is essential because they describe many natural phenomena such as population growth, radioactive decay, and compound interest. At A-Level, you’ll work with the natural exponential function e^x and its inverse, the natural logarithm ln x, and you’ll see how logs can turn multiplication into addition – a property that underpins slide rules and computer algorithms. Mastery of this topic is critical for later topics in calculus and differential equations.

In the wider context of A-Level Mathematics, exponentials and logarithms connect to topics like differentiation and integration, numerical methods, and modelling. They appear frequently in applied units such as mechanics and statistics, so a solid foundation here will pay dividends across the syllabus.

Key Concepts

Core ideas you must understand for this topic

→The definition of a logarithm: if a^x = b, then x = log_a b. Logarithms are the inverse of exponentiation.
→The three laws of logs: log_a (xy) = log_a x + log_a y, log_a (x/y) = log_a x - log_a y, and log_a (x^n) = n log_a x. These are used to simplify expressions and solve equations.
→The natural exponential function e^x and its inverse, the natural logarithm ln x, with special properties: ln e = 1, e^(ln x) = x, and the derivative of e^x is itself.
→Solving exponential equations by taking logs of both sides: if a^x = b, then x = log b / log a (using any base, common log or natural log). Also, equations where both sides can be expressed as powers of the same base.
→Modelling with exponentials using log-linear graphs: reducing non-linear relationships like y = ab^x or y = ax^n to straight-line form for data analysis, estimating parameters from gradient and intercept.

What You Need to Demonstrate

Key skills and knowledge for this topic

Correct use of the laws of logarithms to simplify expressions
Correct application of the inverse relationship between exponential and logarithmic functions
Accurate sketching of y = a^x, y = e^x, and y = ln x graphs
Correct identification of the gradient of e^kx as k*e^kx
Correct solution of equations of the form a^x = b using logarithms

Marking Points

Key points examiners look for in your answers

Correct use of the laws of logarithms to simplify expressions
Correct application of the inverse relationship between exponential and logarithmic functions
Accurate sketching of y = a^x, y = e^x, and y = ln x graphs
Correct identification of the gradient of e^kx as k*e^kx
Correct solution of equations of the form a^x = b using logarithms

Examiner Tips

Expert advice for maximising your marks

💡Always check if the base of the logarithm is specified; assume base 10 if not, but use 'ln' for base 'e'.
💡Use the property that ln(e^x) = x and e^(ln x) = x to simplify complex expressions.
💡When solving a^x = b, taking logs of both sides is the standard approach.
💡Ensure you can sketch the graphs of y = e^x and y = ln x, including their asymptotes and intercepts.
💡Always show the step where you apply a log law explicitly. For example, write 'log(2^x) = x log 2' to secure method marks, even if the final calculation goes awry.
💡When solving exponential equations, first isolate the exponential term if possible (e.g., 3 * 2^x = 12 ⇒ 2^x = 4). Then take logs. This avoids algebraic slips and makes your working clearer.
💡In modelling questions, carefully label your graph axes when plotting log y against x. Use a table to find the log values, and when reading off the intercept, remember it is log a, not a itself, so you must exponentiate to find a.

Common Mistakes

Pitfalls to avoid in your exam answers

Confusing the laws of logarithms (e.g., log(x+y) = log x + log y)
Incorrectly applying the power law for logarithms
Failing to recognize the domain restrictions for logarithmic functions
Errors in algebraic manipulation when solving exponential equations
Students often mistakenly apply log laws to sums or differences, e.g., ln(a + b) = ln a + ln b, which is incorrect. The log laws only apply to multiplication, division, and powers.
Another common error is forgetting that the argument of a logarithm must be strictly positive. When solving log equations, always check for extraneous solutions that make the argument zero or negative.
Confusing ln e with 0 or e^0 with 1: ln e = 1 because e^1 = e, while e^0 = 1. Remember that ln is the power to which e must be raised to get the number.

Revision Plan

How to revise this topic in 1–2 weeks

1Start by reviewing index laws and introducing logarithms as the inverse operation. Watch a short video explanation and complete a diagnostic quiz to check recall of GCSE-level indices.
2Learn and memorise the three log laws. Practise applying them to simplify expressions like log_2 8 + log_2 4 = log_2 32 = 5, and start to use them in simple equations.
3Move on to solving exponential equations: begin with cases that don't require logs (e.g., 3^x = 27 ⇒ x = 3) then progress to using logs for equations like 2^x = 5. Introduce natural logs and the number e through its definition and graph.
4Tackle exponential modelling: learn to reduce y = ab^x to linear form by taking logs, plot log y against x, and interpret the gradient and intercept. Practise with past paper questions that provide data tables.
5Consolidate with mixed exercises covering all subtopics, then complete a timed past paper section focusing on exponentials and logs. Review mistakes and ensure you can handle exam-style questions without notes.

Exam Question Types

How this topic typically appears in the exam

📋‘Solve the equation 5^(2x+1) = 100’ – These questions assess your ability to take logs of both sides and rearrange. Common mistake: forgetting to apply the power rule correctly to the exponent (2x+1).
📋‘Given that y = ab^x, find a and b from the following data…’ – You must take logs (base 10 or e) to form a linear equation, calculate the gradient and intercept from the table, then convert back using exponentials.
📋‘Differentiate f(x) = e^(3x) + 2 ln x’ – Applying standard results: derivative of e^(kx) is ke^(kx) and derivative of ln x is 1/x. These often appear as parts of longer calculus questions.
📋‘Sketch the graphs of y = e^x and y = ln x on the same axes, showing key features’ – Label intercepts: (0,1) for e^x, (1,0) for ln x, and asymptotes (x-axis for e^x, y-axis for ln x). Show they are reflections in y = x.

Frequently Asked Questions

Common questions students ask about this topic

Before You Start

Prior knowledge that will help with this topic

•Fluency with the laws of indices (GCSE Higher tier): a^m × a^n = a^(m+n), (a^m)^n = a^(mn), a^0 = 1, and negative/fractional indices.
•Basic algebraic manipulation, including solving linear equations and using the distributive property, as these are needed when rearranging logarithmic and exponential equations.
•Understanding of functions, their inverses, and graph transformations, particularly reflecting in the line y = x to visualise the relationship between exponential and log graphs.

Key Terminology

Essential terms to know

Laws of logarithms and algebraic manipulation
Exponential growth and decay modelling
Linearisation of non-linear data using logarithmic scales
The natural logarithm and the constant e

Likely Command Words

How questions on this topic are typically asked

Solve

Sketch

Show that

Find

Evaluate

Ready to test yourself?

Practice questions tailored to this topic

F: Exponentials and logarithms

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Revision Plan

Exam Question Types

Frequently Asked Questions

Before You Start

Key Terminology

Likely Command Words

Ready to test yourself?

Related Topics in AQA A-Level Mathematics

A: Proof

B: Algebra and functions

C: Coordinate geometry in the ( x , y ) plane

D: Sequences and series

Topic Synopsis

Key Concepts & Core Principles

Exam Tips & Revision Strategies

Common Misconceptions & Mistakes to Avoid

Examiner Marking Points

F: Exponentials and logarithms

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Revision Plan

Exam Question Types

Frequently Asked Questions

How do I solve an equation like 2^x = 10?

What is the difference between log and ln?

Why is the derivative of ln x equal to 1/x?

How do I rearrange y = ab^x to get a straight-line graph?

When should I use log laws in exam questions?

Before You Start

Key Terminology

Likely Command Words

Ready to test yourself?

Related Topics in AQA A-Level Mathematics

A: Proof

B: Algebra and functions

C: Coordinate geometry in the ( x , y ) plane

D: Sequences and series