The Natural Logarithmic Function: Properties, Derivatives, and Integrals

The Natural Logarithmic Function

Introduction

The natural logarithmic function, denoted as ln(x), is a fundamental concept in calculus. It is defined as the inverse of the exponential function with base e, and has important properties related to differentiation and integration. This study guide covers the definition, properties, derivatives, integrals, and applications of the natural logarithmic function.

Definition of the Natural Logarithm

• Definition: The natural logarithm of x is defined as the area under the curve y = 1/t from t = 1 to t = x:

$\ln x = \int_1^x \frac{1}{t} dt$,   $x > 0$

• Interpretation: ln(x) is an accumulation function representing the area under y = 1/t from 1 to x.

• Sign: ln(x) > 0 if x > 1; ln(x) < 0 if 0 < x < 1.

Derivative of the Natural Logarithm

• Key Formula:

$\frac{d}{dx} [\ln x] = \frac{1}{x}$,   $x > 0$

• This result follows directly from the definition and the First Fundamental Theorem of Calculus.

• For x < 0, $\frac{d}{dx} [\ln|x|] = \frac{1}{x}$.

Examples of Differentiation

• Example 1: Find $\frac{d}{dx} [\ln(2x)]$

$y = \ln(2x)$ $\frac{dy}{dx} = \frac{1}{2x} \cdot 2 = \frac{1}{x}$,   $x > 0$

• Example 2: Find $\frac{d}{dx} [\ln(3x^3 + 14x - 5)]$ and state the domain.

$y = \ln(3x^3 + 14x - 5)$ $\frac{dy}{dx} = \frac{1}{3x^3 + 14x - 5} \cdot (9x^2 + 14)$ Domain: $3x^3 + 14x - 5 > 0$ Critical values: Solve $3x^3 + 14x - 5 = 0$ for $x$.

Integration Involving the Natural Logarithm

• Key Formula:

$\int \frac{1}{x} dx = \ln|x| + C$

• This formula is valid for $x \neq 0$.

• Example: $\int_6^{3x-2} \frac{2}{3x-2} dx = 2 \ln|3x-2| + C$

Properties of Logarithms

• Basic Properties:

Property	Equation
Logarithm of 1	$\ln 1 = 0$
Product Rule	$\ln(ab) = \ln a + \ln b$
Quotient Rule	$\ln\left(\frac{a}{b}\right) = \ln a - \ln b$
Power Rule	$\ln(a^r) = r \ln a$

• Proof of Product Rule: Using the definition and properties of integrals, it can be shown that $\ln(ax) = \ln a + \ln x$ for $a > 0$.

Logarithmic Differentiation

• Technique: Useful for differentiating functions of the form $y = [f(x)]^{g(x)}$ or products/quotients of powers.

• Steps:

  1. Take the natural logarithm of both sides: $\ln y = \ln(f(x))$

  2. Differentiate both sides implicitly.

  3. Solve for $\frac{dy}{dx}$.

• Example: $y = \frac{(2x+3)^3 (3x+2)^2}{x+1}$ $\ln y = 3 \ln(2x+3) + 2 \ln(3x+2) - \ln(x+1)$ Differentiate both sides: $\frac{1}{y} \frac{dy}{dx} = \frac{3}{2x+3} \cdot 2 + \frac{2}{3x+2} \cdot 3 - \frac{1}{x+1}$ Solve for $\frac{dy}{dx}$.

Warnings and Common Mistakes

• When integrating $\int \frac{1}{g(x)} g'(x) dx$, the result is $\ln|g(x)| + C$.

• Do not confuse $\int \frac{1}{y} dy$ with $\ln|y| + C$ if $y$ is a function of $x$.

• Always use absolute value inside the logarithm for indefinite integrals: $\ln|x|$.

Warning	Correct Formula
$\int \frac{1}{g(x)} d(g(x))$	$\ln|g(x)| + C$
$\int \frac{1}{y} d(y)$	$\ln|y| + C$
$\int \frac{1}{y} dy$	$\ln|y| + C$
$\int \frac{1}{x} dx$	$\ln|x| + C$

Summary and Take-Home Message

• The derivative of the natural logarithm is $\frac{1}{x}$ for $x \neq 0$.

• The integral $\int \frac{1}{x} dx$ yields $\ln|x| + C$ for $x \neq 0$.

• Logarithmic properties simplify differentiation and integration of complex expressions.

Additional info: The notes also emphasize the importance of domain restrictions (e.g., $x > 0$ for $\ln x$) and the use of absolute values in integration to account for negative inputs.