Exponentials and Logarithms: Definitions, Properties, and Applications

Exponentials and Logarithms

Introduction to Exponential Functions

Exponential functions are a fundamental concept in calculus, describing quantities that grow or decay at rates proportional to their current value. The general form of an exponential function is:

• Definition: An exponential function is given by $f(x) = a^x$, where a is a positive real number called the base, and x is the exponent or power.

• Example: $2^3 = 8$ means evaluating the function $f(x) = 2^x$ at $x = 3$.

• Domain: Exponential functions are defined for all real numbers $x$.

• Special Cases: The exponent $x$ can be an integer, rational, or even irrational number.

Graphs of Exponential Functions

The graph of $y = a^x$ depends on the value of the base $a$:

• If $a > 1$, the function increases rapidly as $x$ increases.

• If $0 < a < 1$, the function decreases as $x$ increases.

Graphical Features:

• The graph passes through the point $(0, 1)$ for any base $a$.

• For $a > 1$, the graph is increasing; for $a < 1$, it is decreasing.

Graph Comparison Table

Base $a$	Graph Shape	Behavior as $x \to \infty$
$a > 1$	Increasing	$y \to \infty$
$0 < a < 1$	Decreasing	$y \to 0$

Inverse Functions: Logarithms

The inverse of an exponential function is the logarithm. Logarithms allow us to solve for exponents in equations involving exponentials.

• Definition: The logarithm with base $a$ is defined as $F(x) = \log_a(x)$, which is the inverse of $f(x) = a^x$.

• Inverse Relationship:

  • $f^{-1}(x) = \log_a(x)$

  • $a^{\log_a(x)} = x$

  • $\log_a(a^x) = x$

• Notation: $\log_a(x)$ is read as "the logarithm with base $a$ of $x$".

Examples of Exponential and Logarithmic Equations

• Example 1: $f(x) = 3^x$, $F(x) = \log_3(x)$ is its inverse.

  • $3^{\log_3(x)} = x$, for $x > 0$

  • $\log_3(3^x) = x$, for $-\infty < x < \infty$

• Example 2: $3^2 = 9$ means $\log_3(9) = 2$.

• Example 3: $2^{-1/2} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

  • In logarithmic form: $\log_2\left(\frac{\sqrt{2}}{2}\right) = -\frac{1}{2}$

Properties of Logarithms

Logarithms have several important properties that simplify calculations and solve equations:

Property	Formula (LaTeX)
Power Rule	$\log_a(x^r) = r \log_a(x)$
Product Rule	$\log_a(xy) = \log_a(x) + \log_a(y)$
Quotient Rule	$\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$

• Application: These properties allow us to expand, simplify, and solve logarithmic expressions.

• Example: $\log_3(\sqrt{3}) = \log_3(3^{1/2}) = \frac{1}{2} \log_3(3) = \frac{1}{2}$

Euler's Number $e$

Euler's number, denoted $e$, is a fundamental constant in mathematics, especially in calculus and exponential growth/decay problems.

• Definition: $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

• Approximate Value: $e \approx 2.718281828$

• Properties:

  • The function $f(x) = e^x$ is its own derivative.

  • $e^x$ models continuous growth and decay.

Table: Limits Defining $e$

Expression	Limit Value
$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$	$e$
$\lim_{n \to \infty} \left(1 + \frac{k}{n}\right)^n$	$e^k$

Summary of Key Formulas

• Exponential Function: $f(x) = a^x$

• Logarithm (Inverse): $f^{-1}(x) = \log_a(x)$

• Inverse Properties:

  • $a^{\log_a(x)} = x$

  • $\log_a(a^x) = x$

• Logarithm Properties:

  • $\log_a(x^r) = r \log_a(x)$

  • $\log_a(xy) = \log_a(x) + \log_a(y)$

  • $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$

• Euler's Number: $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

Additional info:

• Logarithms are only defined for positive arguments ($x > 0$).

• The natural logarithm, $\ln(x)$, uses base $e$ and is especially important in calculus.

• Exponential and logarithmic functions are used in modeling population growth, radioactive decay, compound interest, and many other real-world phenomena.