Evaluate
log 2 32=?
Since 2 5 =32 , the answer is 5.
🎯 Learning Objectives
- understand a logarithm as an exponent;
- convert between index and logarithmic form;
- evaluate simple logarithms without a calculator.
Introduction
This lesson develops Introduction to Logarithms from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Logs ask an exponent question
A logarithm answers: “What power must the base be raised to, to produce this number?”
2
3
=8
⇔
log
2
8=3
b
x
=N ⇔ log
b
N=x
Conditions: b>0 , b≠1 , and N>0 .
🔄 Index–Log Converter
Enter a base and exponent.
✏️ Worked Examples
Convert
log 5 125=3
Index form: 5 3 =125 .
Common mistake:
In
log
2
8=3
, the base is 2, not 8.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
- Convert 4 3 =64 to logarithmic form.
- Convert log 10 1000=3 to index form.
- Evaluate log 3 81 .
- Evaluate log 7 1 .
Show answers
- log 4 64=3
- 10 3 =1000
- 4
- 0
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.