Natural Log Calculator
Calculate natural logarithm ln(x) with base e (≈ 2.71828)
Must be a positive number
How to Use This Calculator
Enter Number
Input the positive number (x) for which you want to find the natural logarithm. The number must be greater than 0.
Calculate
Click "Calculate ln(x)" to compute the natural logarithm. The result shows what power of e equals your input number.
Verify Result
Check the verification: e raised to the power of the result should equal your input number.
Formula
ln(x) = loge(x)
Where:
- x = positive number
- e = Euler's number ≈ 2.718281828459045...
- ln(x) = natural logarithm of x
- If ey = x, then ln(x) = y
Example 1: ln(e)
ey = e
e¹ = e
Therefore: ln(e) = 1
Example 2: ln(1)
ey = 1
e⁰ = 1
Therefore: ln(1) = 0
Example 3: ln(e²)
ey = e²
e² = e²
Therefore: ln(e²) = 2
Example 4: ln(10)
ln(10) ≈ 2.302585
e2.302585 ≈ 10
About Natural Log Calculator
The Natural Log Calculator computes the natural logarithm (ln) of a number. The natural logarithm uses Euler's number e (≈ 2.71828) as its base. Natural logarithms are fundamental in calculus, especially because the derivative of ln(x) is 1/x, making it essential in differentiation and integration. Natural logs appear frequently in exponential growth models, continuous compound interest, and many scientific applications.
When to Use This Calculator
- Calculus: Calculate natural logarithms for differentiation and integration
- Continuous Compound Interest: Calculate investments with continuous compounding
- Exponential Growth/Decay: Model population growth, radioactive decay, or bacterial growth
- Statistics: Calculate probabilities in log-normal distributions
- Physics: Solve problems involving exponential decay, RC circuits, or wave functions
Why Use Our Calculator?
- ✅ Instant Results: Get accurate natural logarithm calculations immediately
- ✅ High Precision: Uses JavaScript's Math.log() for maximum accuracy
- ✅ Verification: Shows verification that e^result equals your input
- ✅ Educational: Learn about natural logarithms and their applications
- ✅ 100% Free: No registration or payment required
- ✅ Accurate: High precision calculations
Common Applications
Continuous Compound Interest: If you invest $1,000 at 5% annual interest compounded continuously for 10 years: A = 1000 × e0.05×10 = 1000 × e0.5 ≈ $1,648.72.
Radioactive Decay: The amount of radioactive material after time t is N(t) = N₀ × e-λt, where λ is the decay constant. Taking ln helps solve for t.
Calculus: The derivative of ln(x) is 1/x, and the integral of 1/x is ln|x| + C. This makes natural logarithms essential in calculus.
Tips for Best Results
- The input number must be positive
- ln(1) = 0 (any logarithm of 1 is 0)
- ln(e) = 1 (logarithm of the base always equals 1)
- ln(en) = n (logarithm and exponent cancel)
- For approximate conversions: ln(x) ≈ 2.303 × log₁₀(x)
Frequently Asked Questions
What is natural logarithm?
Natural logarithm (ln) is the logarithm with base e (Euler's number ≈ 2.71828). It's written as ln(x) or loge(x) and is fundamental in calculus and many scientific applications.
What's the difference between ln and log?
ln uses base e (≈ 2.71828), while log usually means base 10. Both follow the same rules, just different bases. ln(x) = loge(x), while log(x) typically means log₁₀(x).
Why is it called "natural" logarithm?
It's called natural because it arises naturally in calculus - the derivative of ln(x) is 1/x, which is simpler than other bases. It also appears naturally in exponential growth and many physical processes.
How do I convert ln to log base 10?
Use the change of base formula: ln(x) = log₁₀(x) / log₁₀(e) ≈ log₁₀(x) / 0.4343. Alternatively, ln(x) ≈ 2.303 × log₁₀(x).
What is ln(e)?
ln(e) = 1. This is because e¹ = e, and ln is the inverse of ex. The logarithm of the base always equals 1: logb(b) = 1 for any base b.