Product law
log
b
(MN)=log
b
M+log
b
N
🎯 Learning Objectives
- expand and condense logarithmic expressions;
- apply product, quotient and power laws;
- recognise invalid log-law shortcuts.
Introduction
This lesson develops Laws of 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.
Visual Exploration
Notice→Connect→Explain
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
The Laws of Logarithms
Quotient law
log
b
(M/N)=log
b
M−log
b
N
Power law
log
b
(M
k
)=k log
b
M
Why?
The laws follow from the corresponding laws of indices.
✏️ Expand
Expand log 2 (8x 3 /y) .
Use the quotient law:
log
2
(8x
3
)−log
2
y
.
Use the product law:
log
2
8+log
2
(x
3
)−log
2
y
.
Use the power law and evaluate
log
2
8=3
.
Answer:
3+3log
2
x−log
2
y
.
Condensing
2log 3 x+log 3 y−log 3 z = log 3 (x 2 y/z)
Never write:
log(M+N)=log M+log N
. There is no addition law for logarithms.
🧠 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
- Expand log 5 (25x) .
- Expand log 2 (x 4 /y) .
- Condense log 3 a+2log 3 b .
Show answers
- 2+log 5 x
- 4log 2 x−log 2 y
- log 3 (ab 2 )
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.