Exponential and Logarithmic Functions: Definitions, Properties, and Applications

Exponentials and Logarithms

Introduction to Exponential Functions

Exponential functions are a fundamental class of functions in calculus, characterized by a constant base raised to a variable exponent.

• Exponential Function: A function of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Base (a): The constant that is raised to a power.

• Exponent (x): The variable in the exponent position.

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

• When $a > 1$, the function is increasing; when $0 < a < 1$, the function is decreasing.

Example: $2^3 = 8$ means the value of the exponential function $f(x) = 2^x$ at $x = 3$ is 8.

Graphs of Exponential Functions

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

• For $a > 1$, the graph increases rapidly as $x$ increases.

• For $0 < a < 1$, the graph decreases as $x$ increases.

• All exponential graphs pass through the point $(0, 1)$.

Additional info: The horizontal asymptote for all exponential functions is $y = 0$.

Inverse Functions: Logarithms

Exponential functions have inverses called logarithmic functions. The logarithm with base $a$ is the inverse of the exponential function with base $a$.

• Logarithm: $\log_a(x)$ is defined as the exponent to which $a$ must be raised to obtain $x$.

• Formally, $f(x) = a^x$ and $f^{-1}(x) = \log_a(x)$.

• By definition: $a^{\log_a(x)} = x$ for $x > 0$ and $\log_a(a^x) = x$ for all real $x$.

Example: $3^{\log_3(x)} = x$ for $x > 0$; $\log_3(3^x) = x$ for all $x$.

Relationship Between Exponentials and Logarithms

• The logarithm and exponential functions are inverses of each other.

• General relationship: $a^{\log_a(Q)} = Q$ for $Q > 0$.

• Logarithmic equations can be rewritten as exponential equations and vice versa.

Example: $\log_3(9) = 2$ is equivalent to $3^2 = 9$.

Properties of Logarithms

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

Property	Formula
Power Rule	$\log_a(Q^\Delta) = \Delta \log_a(Q)$
Product Rule	$\log_a(Q\Delta) = \log_a(Q) + \log_a(\Delta)$
Quotient Rule	$\log_a\left(\frac{Q}{\Delta}\right) = \log_a(Q) - \log_a(\Delta)$

• These properties hold for all $Q > 0$, $\Delta > 0$, and $a > 0$, $a \neq 1$.

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

Examples and Applications

• Example 1: Write $2^{-1/2} = \frac{\sqrt{2}}{2}$ as a logarithm.

  • $2^{-1/2} = \frac{\sqrt{2}}{2}$

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

• Example 2: Evaluate $\log_3(\sqrt{3})$ exactly.

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

Euler's Number and the Natural Exponential Function

Definition of Euler's Number ($e$)

Euler's number, denoted $e$, is a fundamental mathematical constant approximately equal to 2.718281828. It is the base of the natural logarithm (ln).

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

• This limit defines $e$ as the value approached by the expression as $n$ becomes very large.

• More generally, $\lim_{n \to \infty} \left(1 + \frac{b}{n}\right)^n = e^b$

Additional info: The function $f(x) = e^x$ is called the natural exponential function, and its inverse is the natural logarithm $\ln(x)$.

Properties of Sequences Leading to $e$

• The sequence $a_n = \left(1 + \frac{1}{n}\right)^n$ is increasing and bounded above by 3, so it converges to $e$.

• If a sequence is increasing and bounded, it converges to a finite limit.

• If a sequence is not bounded above, it diverges to $+\infty$.

Applications of $e$

• $e$ appears in continuous compound interest, population growth, and calculus (especially in differentiation and integration of exponential functions).

• For any real number $b$, $\lim_{n \to \infty} \left(1 + \frac{b}{n}\right)^n = e^b$.

Summary Table: Exponential and Logarithmic Relationships

Exponential Form	Logarithmic Form
$a^x = Q$	$\log_a(Q) = x$
$e^x = Q$	$\ln(Q) = x$

Additional info: The natural logarithm $\ln(x)$ is shorthand for $\log_e(x)$, and is widely used in calculus due to its simple derivative and integral properties.