Resources:
- Book Calculus by Thomas/Finney, 9th Edition, Page 458
- C. to, “logarithm to the base of the mathematical constant e,” Wikipedia.org, Jul. 20, 2001. https://en.wikipedia.org/wiki/Natural_logarithm (accessed Feb. 01, 2026).
What is Natural Logarithm?
In calculus, the natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.71828.
Unlike base-10 logarithms (common logs), the natural logarithm, denoted as $ln(x)$, is preferred in calculus because it simplifies the differentiation and integration of exponential functions.
1. Defining the Natural Logarithm
In calculus, $ln(x)$ is often defined as the area under the curve of the reciprocal function $f(t)=\frac{1}{t}$ from $1$ to $x$:
This definition is crucial because it links the logarithm directly to integration, showing that the derivative of $ln(x)$ is $\frac{1}{x}$.
2. Key Properties
The natural logarithm follows the standard laws of logarithms, which are frequently used to simplify complex functions before differentiating them:
- Product Rule: $ln(ab)=ln(a)+ln(b)$
- Quotient Rule: $ln(\frac{a}{b})=ln(a)−ln(b)$
- Power Rule: $ln(a^n)=n ln(a)$
- Special Values: $ln(1)=0$ and $ln(e)=1$
3. Calculus Operations
The “natural” part of the name comes from how cleanly it behaves in operations.
Differentiation
The derivative of $\ln(x)$ is the simplest of any logarithm: \[ \frac{d}{dx} \ln(x) = \frac{1}{x} \]
https://www.geeksforgeeks.org/maths/derivative-of-ln-x/
If you have a function $u(x)$ inside the log, you use the Chain Rule: \[ \frac{d}{dx} \ln(u) = \frac{1}{u} \cdot \frac{du}{dx} \]
Integration
While $\frac{1}{x}$ integrates to $\ln |x|$, the integral of $\ln(x)$ itself requires Integration by Parts: \[ \int \ln(x) \, dx = x \ln(x) – x + C \]
http://math2.org/math/integrals/more/ln.htm