Indices and Logarithms - Mathematics (Senior Secondary)

This course provides a comprehensive guide to indices and their inverse, logarithms. It treats them not as separate topics, but as two sides of the same coin. The course begins with a mastery of the laws of indices before using that knowledge to define and derive the principles of logarithms, which are essential tools for simplifying complex calculations. A command of indices and logarithms is non-negotiable for success in any advanced science, technology, engineering, or mathematics (STEM) field. These concepts are fundamental to modelling exponential growth and decay, analysing data on logarithmic scales (like pH or decibels), and solving advanced equations in calculus and physics. By the end of this course, you will be able to confidently apply the laws of indices to simplify expressions and solve equations. You will understand the definition of a logarithm, read logarithm and antilogarithm tables, and use them to perform complex calculations involving multiplication, division, powers, and roots. This course is built for students preparing for higher-level mathematics. It is also essential for anyone entering scientific or financial fields who needs a strong command of these foundational calculation tools. A solid understanding of basic algebra is required.

Enrolment valid for 12 months

Course Chapters

1. Introduction

This chapter introduces the core focus of the course. We will define indices and logarithms and immediately establish the fundamental inverse relationship between them. By the end of this chapter, you will understand the course structure. You will be able to explain how logarithms are directly connected to indices, setting the stage for all future lessons.

Chapter lessons

1-1. Welcome

This lesson outlines the course objectives and structure. It introduces the two main topics and prepares you for a unified approach to learning them.

1-2. Understanding the link between indices and logarithms

This lesson establishes the critical inverse relationship between exponentiation and logarithms. This core concept is the foundation upon which the entire course is built.

2. Working with Indices

This chapter is dedicated to achieving proficiency with indices. We will review standard form and then conduct a rigorous study of the laws of indices and their application. By the end of this chapter, you will be able to recall and apply all the laws of indices. You will use them efficiently to simplify complex expressions and solve indicial equations.

Chapter lessons

2-1. Revision of standard form

This lesson revisits the concept of standard form (scientific notation). This serves as a practical starting point for discussing powers of ten.

2-2. The laws of indices

This lesson provides a systematic breakdown of the laws of indices. We will cover the rules for multiplication, division, powers, the zero index, and negative indices.

2-3. Applying the laws of indices to problem solving

This lesson moves from theory to practice. You will solve a variety of problems using the laws of indices to manipulate and simplify algebraic expressions.

3. Understanding Logarithms

This chapter uses your knowledge of indices to build a solid conceptual understanding of logarithms. We will formally define a logarithm and derive its laws directly from the laws of indices. By the end of this chapter, you will be able to define a logarithm as the inverse of an index. You will understand the laws of logarithms and interpret logarithmic graphs.

Chapter lessons

3-1. Definition of a logarithm

This lesson formally defines the logarithm. We will use simple examples to show that finding a logarithm is the same as finding an unknown power.

3-2. Deducing logarithms from indices

This lesson demonstrates how the laws of logarithms are not new rules to memorise. They are the laws of indices expressed in a different format.

3-3. The graph of y = 10^x

This lesson explores the exponential function. We will analyse its graph to visually understand the concept of exponential growth and its link to logarithms.

4. Using Logarithm Tables

This chapter focuses on the practical application of logarithms as a computational tool. You will learn the mechanical skills of reading logarithm and antilogarithm tables to solve complex arithmetic problems. By the end of this chapter, you will be able to read logarithm and antilogarithm tables accurately. You will use these tables to perform calculations that would be tedious by hand.

Chapter lessons

4-1. Reading logarithm and antilogarithm tables

This lesson teaches the technical skill of using four-figure tables. You will learn how to find the logarithm of a number and reverse the process to find the antilogarithm.

4-2. Effective use of tables in calculations

This lesson applies your table-reading skills. You will solve problems involving multiplication, division, powers, and roots using the principles of logarithms.

5. Conclusion and Next Steps

This final chapter summarises the key skills covered in the course. It reinforces the connection between indices and logarithms and prepares you for more advanced topics. By the end of this chapter, you will have reviewed the laws and applications of indices and logarithms. You will understand how these tools are used in higher-level mathematics.

Chapter lessons

5-1. Summary of what you have learned

This lesson provides a full review of the course content. We will recap the laws of indices, the definition of logarithms, and their use as a calculation tool.

5-2. Preparing for exponential functions and advanced algebra

This lesson looks ahead to future topics. We will explain how a strong command of indices and logarithms is a prerequisite for studying functions, calculus, and advanced algebra.