Natural Logarithm (ln) Calculator
What is a Natural Logarithm (ln)?
The natural logarithm (ln) is a mathematical function that answers the question:
“To what power must we raise the number e (Euler’s number ≈ 2.71828) to get a given value?”
Mathematically:
If ln(x) = y, then eʸ = x.
Key Properties:
✔ ln(1) = 0 (because e⁰ = 1)
✔ ln(e) = 1 (because e¹ = e)
✔ ln(x) is only defined for x > 0
✔ Used in calculus, compound interest, physics, and algorithms
User Guide for Natural Logarithms Calculator
How to Use the Calculator
- Enter a Positive Number
- Type any positive number (e.g.,
2,10,3.14, or0.5). - ❌ Negative numbers & zero won’t work (ln is undefined).
- Click “Calculate Natural Logarithm”
- The tool computes ln(x) using JavaScript’s
Math.log().
- View Results
- Exact ln value (up to 6 decimal places).
- Step-by-step explanation of the calculation.
Example Calculations
| Input (x) | ln(x) ≈ | Explanation |
|---|---|---|
| 1 | 0 | Because e⁰ = 1 |
| 2.71828 | 1 | Because e¹ ≈ e |
| 10 | 2.302585 | e²·³⁰²⁵⁸⁵ ≈ 10 |
| 0.5 | -0.693147 | e⁻⁰·⁶⁹³¹⁴⁷ ≈ 0.5 |
Common Applications
🔹 Compound Interest: Calculating continuous growth rates.
🔹 Calculus: Solving integrals & derivatives.
🔹 Algorithms: Measuring time complexity (e.g., binary search).
🔹 Physics: Describing exponential decay (e.g., radioactive half-life).
FAQs
Q: Why does ln(0) give an error?
A: Because eʸ never equals 0 for any real number y!
Q: What’s the difference between log(x) and ln(x)?
A:
- log(x) = log₁₀(x) (base 10)
- ln(x) = logₑ(x) (base e)
Q: Can I compute ln(-5)?
A: No, but in complex numbers, ln(-x) = ln(x) + πi.