Indices, surds and logarithms are three views of one idea: what happens when a number is raised to a power. This page shows how they connect, the order to study them in and where to start.
How do the three topics connect?
An index tells you a number multiplied by itself a given number of times. A surd is an irrational root that cannot be written as a fraction, such as √2, and a root is a fractional index: √2 = 2^(1/2). A logarithm asks the reverse question: which power gives this number?
Here is one line that shows the link. The statement 2³ = 8 and the statement log₂ 8 = 3 say exactly the same thing, written two ways.
Because of this link, it is easy to mix the laws of one topic into another. The four lessons below keep each rule in its own place.
What order should I study them in?
- Applying index laws with fractional powers gives you the base skill.
- Simplifying and rationalising surds applies it to roots.
- Using logarithm laws with domain checks turns the index laws around.
- Solving exponential and logarithmic equations puts all three to work.
Finish with the chapter practice set, which mixes the four skills.
Who should start where?
A student who writes 8^(2/3) as 8 × 2 ÷ 3 should start with the index lesson. A student who is sure about indices but loses marks on roots should go to surds.
A student whose logarithm answers are right on paper but rejected in the marking, because a value makes the log undefined, should start with the domain lesson. A student with little time should do the practice set first and return to the lesson for any question that fails.
If you would like a teacher to watch your working, see online one-to-one Additional Mathematics tuition. The wider SPM Additional Mathematics guide shows how this chapter fits with the rest.