Logarithms
The inverse of exponentiation.
Key idea
The inverse of exponentiation.
In depth
log_b(x) = y means bʸ = x. A logarithm asks what power of b gives x. It turns multiplication into addition, which tames huge ranges (decibels, pH, earthquakes).
Worked example
log₂(8) = 3 since 2³ = 8. log₁₀(1000) = 3.
Formula
Real-world uses
- Measuring earthquake magnitudes, sound levels in decibels and pH in chemistry
Common mistake
Trying to take the log of zero or a negative number; the domain is x > 0.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Logarithms?
The inverse of exponentiation. It is a topic at the advanced level in Algebra.
What is the formula for Logarithms?
The key formula is log_b(x) = y ⇔ bʸ = x.
Where is Logarithms used in real life?
Real-world examples: Measuring earthquake magnitudes, sound levels in decibels and pH in chemistry.
What should I learn before Logarithms?
Make sure you are comfortable with The Number e, the previous topic in Algebra.
What is a common mistake with Logarithms?
Trying to take the log of zero or a negative number; the domain is x > 0.
How can I practise Logarithms?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Logarithms?
Move on to Laws of Logarithms next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.