🎯 Learning Objectives
- solve equations containing a single logarithm;
- combine logarithms before converting to index form;
- check that solutions satisfy log-domain restrictions.
Introduction
This lesson develops Solving Logarithmic Equations 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
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
✏️ Single logarithm
Solve log 3 (x+1)=2 .
✏️ Combine first
Solve log 2 x+log 2 (x−2)=3 .
Domain Check
Every logarithm argument must be greater than zero. Reject any candidate that makes an argument zero or negative.
🧠 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
- Solve log 5 x=3 .
- Solve log 2 (x−1)=4 .
- Solve log 3 x+log 3 (x−2)=1 .
Show answers
- x=125
- x=17
- x=3 ; the other quadratic root is invalid.
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.