Integral Form of the Definition of Natural Logarithm ln(x)

Explore the definition of the natural logarithm as an area under the curve using Simpson's rule for numerical integration.

\[ \ln(x) = \int_1^x \dfrac{1}{t} \, dt , \quad x \gt 0 \]

The integral is computed numerically using Simpson's rule with adjustable precision.

Understanding the Definition

The natural logarithm function, ln(x), is defined as the area under the curve of the function f(t) = 1/t from t = 1 to t = x.

Key Properties:

  • ln(1) = 0: When x = 1, the integral has zero width
  • ln(x) > 0 for x > 1: Area is positive when x > 1
  • ln(x) < 0 for 0 < x < 1: The integral direction reverses
  • ln(x) is the inverse of eˣ: They undo each other

Simpson's Rule

This numerical integration method approximates the area under a curve by dividing it into parabolic segments. It's more accurate than the trapezoidal rule for smooth functions like 1/t.

The rule requires an even number of intervals (n) and provides excellent accuracy with relatively few subdivisions.

Interactive Calculator

Adjust the parameters below to explore how the integral definition works.

Value of x: 2.0
0.1 5.0 10.0
Number of intervals (n): 20
2 100 200

Note: n must be even for Simpson's rule (automatically adjusted if odd)

Results

Approximation using Simpson's rule: 0.693147
Exact value from Math.log(): 0.693147
Absolute difference: 0.000000
Relative error: 0.0000%

Integral Representation:

\[ \int_1^{2.0} \frac{1}{t} \, dt \approx 0.693147 \]

Mathematical Insights

Why This Definition?

This integral definition is fundamental because it:

  • Provides a geometric interpretation
  • Naturally leads to the derivative d/dx ln(x) = 1/x
  • Connects logarithms to calculus concepts
  • Works for all positive real numbers

Accuracy Notes

Simpson's rule error decreases as ~1/n⁴. This means doubling n reduces error by about 16×.

For Better Accuracy:

  • Increase n for larger x values
  • Use n ≥ 20 for x ≤ 5
  • Use n ≥ 50 for x ≤ 10
  • Try different n values to see convergence

Current Calculation:

x = 2.0

n = 20 intervals

Interval width = 0.05

Step-by-Step Tutorial

  1. The calculator starts with x = 2. Click "Calculate ln(x)" to compute ln(2) using Simpson's rule.
  2. Use the x-slider to change the value. Watch how the integral value changes with x.
  3. Adjust the n-slider to change precision. Higher n gives better accuracy but slower calculation.
  4. Try the preset examples: ln(2), ln(3), and ln(0.5) to see different cases.
  5. Compare the Simpson's rule approximation with the exact value from Math.log().
  6. Notice how the error decreases as you increase n (number of intervals).
  7. Explore x values between 0.1 and 10 to understand ln(x)'s full behavior.
  8. Reset to default values at any time using the reset button.

Quick Tip

For x close to 1, ln(x) ≈ x - 1. This linear approximation helps explain why the integral definition makes sense geometrically.