Natural Logarithm
Log with base e.
Key idea
Log with base e.
In depth
ln x is the logarithm base e. It undoes eˣ: ln(eˣ) = x. It is the natural choice in calculus because d/dx ln x = 1/x.
Worked example
ln(e³) = 3. e^(ln 5) = 5. Solve eˣ = 7 → x = ln 7 ≈ 1.946.
Formula
Real-world uses
- Calculating continuous growth rates in finance, biology and population studies
Common mistake
Using ln when you need log₁₀ in formulas that specify base ten.
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 Natural Logarithm?
Log with base e. It is a topic at the advanced level in Algebra.
What is the formula for Natural Logarithm?
The key formula is ln(eˣ) = x.
Where is Natural Logarithm used in real life?
Real-world examples: Calculating continuous growth rates in finance, biology and population studies.
What should I learn before Natural Logarithm?
Make sure you are comfortable with Exponential and Log Equations, the previous topic in Algebra.
What is a common mistake with Natural Logarithm?
Using ln when you need log₁₀ in formulas that specify base ten.
How can I practise Natural Logarithm?
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 Natural Logarithm?
Move on to Arithmetic Series 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.