The Natural Logarithm as an Integral

Definition

The natural logarithm $\ln x$ is defined as the definite integral of $1/t$ from $1$ to $x$ for $x > 0$: $\ln x = \int_{1}^{x} \frac{1}{t} \, dt$

• How to read: “The natural logarithm of x is equal to the integral from one to x of one divided by t with respect to t.”
• Meaning: $\ln x$ is the signed area under $y = 1/t$ from $1$ to $x$. This integral definition is the foundation for all logarithm properties and for defining $e$.

Why It Matters

Defining $\ln x$ as the area under $1/t$ provides the rigorous foundation for all logarithmic calculations in calculus. Without this definition, the connection between power rules and logarithms is lost, making it impossible to evaluate integrals that result in logarithmic growth.

Core Concepts

• Area under 1/t: Geometrically, $\ln x$ represents the net area under the curve $y = 1/t$ between $1$ and $x$.

• Fundamental Theorem of calculus: By the FTC, the derivative of $\ln x$ is $\frac{1}{x}$, meaning $\ln x$ is the antiderivative of $1/x$ that is zero at $x=1$.

  • How to read: “The derivative with respect to x of the natural logarithm of x is equal to one divided by x.”
  • Meaning / when to use: Confirms $\int \frac{1}{x}\,dx = \ln|x| + C$ — the power rule’s missing case $n = -1$. The anchor condition $\ln(1) = 0$ fixes the constant of integration.

• The Number e: The mathematical constant $e$ is defined as the unique number such that $\ln e = 1$.

  • How to read: “The natural logarithm of e is equal to one.”
  • Meaning: $e$ is the input whose area-under-$1/t$ from $1$ to $e$ is exactly one unit — the base of natural growth and the inverse partner of $\ln x$.

Connected Concepts

• The Definite Integral
• Natural Logarithm Properties
• Function Definition, Function Notation