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 ratio of integers. Every real number can be approximated arbitrarily closely by a rational number.

• Rational numbers: Numbers expressible as $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$.

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

• Set notation: $\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

Definition of Euler’s Exponential Function

The exponential function with base $e$ is fundamental in calculus and analysis. It is defined for all real numbers and has unique properties.

• 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 rational ($q > 0$): $e^x = e^{\frac{p}{q}} = \sqrt[q]{e^p}$

• If $x$ is irrational: $e^x = \lim_{n \to \infty} e^{x_n}$, where $x_n$ is a sequence of rationals converging to $x$

Graph of the Exponential Function

The graph of $y = e^x$ is always increasing, passes through $(0,1)$, and approaches zero as $x \to -\infty$.

• Key features:

  • Domain: $(-\infty, \infty)$

  • Range: $(0, \infty)$

  • Horizontal asymptote: $y = 0$ 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 $x, y$

• Identity: $e^0 = 1$

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

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

• 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:

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

• Using the series expansion, this limit evaluates to $e^x$

The Natural Logarithm: Definition and Notation

The natural logarithm is the inverse function of the exponential function $e^x$. It is denoted as $\ln(x)$, $\log_e(x)$, or simply $\log(x)$ in some contexts.

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

• Domain: $x > 0$

• Range: $(-\infty, \infty)$

Table: Notation for the Natural Logarithm

Expression	Meaning
$\ln(x)$	Natural logarithm of $x$
$\log_e(x)$	Logarithm base $e$
$\log(x)$	Logarithm (often base $e$ in calculus)

Properties of the Natural Logarithm and Exponential Functions

• $\ln(1) = 0$

• $\ln(e) = 1$

• $\ln(ab) = \ln(a) + \ln(b)$ for $a, b > 0$

• $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$ for $a, b > 0$

• $e^{\ln(x)} = x$ for $x > 0$

• $\ln(e^x) = x$ for any $x$

General Exponential Functions

Any exponential function with base $a > 0$ can be written in terms of $e$:

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

Derivative of the Natural Logarithm

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

• $\frac{d}{dx} \ln x = \frac{1}{x}$

• More generally, for $u(x)$ differentiable and $u(x) > 0$:

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

Example: Derivative of $\ln(u(x))$

• Let $u(x) = x^2 + 2x + 1$

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

Application: Hyperbolic Cosine Function

The hyperbolic cosine function is defined as:

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

• It is important in physics and engineering, such as describing the shape of a hanging chain (catenary).

• General form: $g(x) = \frac{e^{ax} + e^{-ax}}{2}$, $a > 0$

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

Property	$e^x$	$\ln(x)$
Domain	$(-\infty, \infty)$	$(0, \infty)$
Range	$(0, \infty)$	$(-\infty, \infty)$
Derivative	$e^x$	$\frac{1}{x}$
Inverse	$\ln(x)$	$e^x$
Key Identity	$e^{x+y} = e^x e^y$	$\ln(ab) = \ln(a) + \ln(b)$

Additional info:

• The notes also briefly mention the use of exponential and logarithmic functions in applications such as physics (radioactive decay, catenary curves).

• For more rigorous proofs and deeper theory, consult a textbook on Real Analysis.