Euler’s Exponential Function and the Natural Logarithm: Definitions, Properties, and Calculus Applications

Euler’s Exponential Function and the Natural Logarithm

Introduction to Real Numbers and Rational Approximations

The set of real numbers (ℝ) includes both rational and irrational numbers. Rational numbers (ℚ) are those that can be expressed as fractions, while irrational numbers cannot be written as a simple fraction. Every real number can be approximated arbitrarily closely by a rational number.

• Rational numbers: Numbers that can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

• Irrational numbers: Numbers that cannot be written as a fraction, e.g., $e \approx 2.71$, $\pi \approx 3.14$.

• Set notation: $\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$, where $\mathbb{Z}$ is the set of integers.

Definition of Euler’s Exponential Function

The exponential function with base $e$ is fundamental in calculus and analysis. It is defined differently depending on whether the exponent is an integer, rational, or irrational.

• If $x = p$ is an integer:

  • $e^x = e^p = \underbrace{e \cdot e \cdot \ldots \cdot e}_{p \text{ times}}$

• If $x = \frac{p}{q}$ is a rational number ($q > 0$):

  • $e^x = e^{\frac{p}{q}} = \sqrt[q]{e^p}$

• If $x$ is irrational:

  • Let $x_n = \frac{p_n}{q_n}$ be a sequence of rational numbers converging to $x$.

  • $e^x = \lim_{n \to \infty} e^{x_n}$

Graph of the Exponential Function

The graph of $y = e^x$ is an increasing, convex curve that passes through $(0,1)$ and grows rapidly for large $x$.

• Key features: Always positive, never touches the $x$-axis, and increases without bound as $x \to \infty$.

Properties of $e^x$

The exponential function $e^x$ has several important algebraic and calculus properties.

• Exponent addition: $e^{x+y} = e^x e^y$ for any numbers $x, y$; $e^0 = 1$

• Taylor series expansion: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$

• Exponent subtraction: $e^{x-y} = \frac{e^x}{e^y}$

• Limits:

  • $\lim_{x \to \infty} e^x = +\infty$

  • $\lim_{x \to -\infty} e^x = 0$

• Derivative: $\frac{d}{dx} e^x = e^x$

Proof of the Derivative of $e^x$

The derivative of $e^x$ can be shown using the definition of the derivative and the Taylor series expansion.

• Derivative definition:

  • $\lim_{h \to 0} \frac{e^{x+h} - e^x}{h} = e^x \lim_{h \to 0} \frac{e^h - 1}{h}$

  • Using the Taylor series for $e^h$:

  • $e^h = 1 + h + \frac{h^2}{2!} + \frac{h^3}{3!} + \cdots$

  • So, $\frac{e^h - 1}{h} = 1 + \frac{h}{2!} + \frac{h^2}{3!} + \cdots \to 1$ as $h \to 0$

  • Therefore, $\frac{d}{dx} e^x = e^x$

The Natural Logarithm: Definition and Notation

The inverse of the exponential function is called the natural logarithm. It is denoted by $\ln(x)$, $\log_e(x)$, or simply $\log(x)$ in calculus.

• Definition: If $y = e^x$, then $x = \ln(y)$.

• Notation: $\ln(x)$ is the most common symbol in calculus.

Properties of the Natural Logarithm

The natural logarithm has several key properties that are useful in algebra and calculus.

Property	Equation	Condition
Logarithm of 1	$\ln(1) = 0$
Logarithm of $e$	$\ln(e) = 1$
Logarithm of a product	$\ln(AB) = \ln(A) + \ln(B)$	$A, B > 0$
Logarithm of a quotient	$\ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B)$	$A, B > 0$
Logarithm of a power	$\ln(A^n) = n \ln(A)$	$A > 0$

Application: Hyperbolic Cosine Function

The hyperbolic cosine function is defined using the exponential function and is important in physics and engineering.

• Definition: $\cosh(x) = \frac{e^x + e^{-x}}{2}$

• Example: Find the derivative of $f(x) = \frac{e^x + e^{-x}}{2}$ at $x = 2$.

• Application: The function $g(x) = \frac{e^{ax} + e^{-ax}}{2}$, $a > 0$, describes the shape of a hanging chain (catenary curve).

General Exponential Functions

Any exponential function with base $a > 0$ can be defined using Euler’s exponential function.

• Definition: $a^x = e^{x \ln a}$ for any real $x$

• Calculator note: Many calculators use this definition to compute powers for arbitrary bases.

Derivative of the Natural Logarithm

The derivative of the natural logarithm function is a fundamental result in calculus.

• Derivative of $\ln x$:

  • $\frac{d}{dx} \ln x = \frac{1}{x}$

• Chain Rule for $\ln u(x)$:

  • $\frac{d}{dx} \ln u(x) = \frac{1}{u(x)} \cdot u'(x)$

  • Or, $\frac{d}{dx} \ln u(x) = \frac{u'(x)}{u(x)}$

• Example: For $u(x) = x^2 + 2x + 1$,

  • $\frac{d}{dx} \ln u(x) = \frac{2x + 2}{x^2 + 2x + 1}$

Summary Table: Key Properties of $e^x$ and $\ln x$

Function	Derivative	Key Property
$e^x$	$e^x$	Exponential growth
$\ln x$	$\frac{1}{x}$	Inverse of $e^x$
$a^x$	$a^x \ln a$	General exponential

Additional info:

• The notes provide foundational calculus concepts for exponential and logarithmic functions, including their definitions, properties, and differentiation rules.

• Applications such as the hyperbolic cosine and catenary curve are relevant in physics and engineering.