Exponential and Logarithmic Functions: Definitions, Properties, and Applications

Rational Functions

Definition and Domain

A rational function is any function that can be written as the ratio of two polynomial functions, that is, $h(x) = \frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are polynomials and $g(x) \neq 0$.

• Domain: The set of all real numbers except those for which $g(x) = 0$ (i.e., where the denominator is zero).

• Example: For $h(x) = \frac{x-1}{x-2}$, the domain is $(-\infty, 2) \cup (2, \infty)$ because $x = 2$ makes the denominator zero.

Additional info: Rational functions are important in calculus for understanding discontinuities and asymptotic behavior, which are explored further in graphing and limits.

Exponential Functions

Definition and Properties

An exponential function is a function of the form $f(x) = b^x$, where the base $b$ is a positive real number not equal to 1 ($b > 0$, $b \neq 1$).

• Range: $b^x > 0$ for all real $x$; the function never takes zero or negative values.

• Behavior:

  • If $b > 1$, $b^x$ increases as $x$ increases (exponential growth).

  • If $0 < b < 1$, $b^x$ decreases as $x$ increases (exponential decay).

• Examples:

  • For $b = 2$: $2^0 = 1$, $2^1 = 2$, $2^2 = 4$, $2^{-1} = \frac{1}{2}$, $2^{-2} = \frac{1}{4}$, etc.

  • For $b = \frac{1}{2}$: $(\frac{1}{2})^0 = 1$, $(\frac{1}{2})^1 = \frac{1}{2}$, $(\frac{1}{2})^2 = \frac{1}{4}$, $(\frac{1}{2})^{-1} = 2$, etc.

The Natural Exponential Function and the Number e

Definition and Origin of e

The natural exponential function is $f(x) = e^x$, where $e \approx 2.71828$ is an irrational and transcendental number. The number $e$ arises naturally in the context of continuous growth and compound interest.

• Compound Interest Example: If $1 is invested at 100% interest compounded$n$times per year, the amount after one year is$(1 + \frac{1}{n})^n$.

• As $n \to \infty$, $(1 + \frac{1}{n})^n \to e$.

• General Formula: $e = \lim_{n \to \infty} (1 + \frac{1}{n})^n$

Additional info: The function $e^x$ is unique in that its rate of change (derivative) is equal to itself, a property fundamental to calculus.

Exponential Models

General Form and Applications

Exponential models are used to describe processes that grow or decay at rates proportional to their current value. The general form is $P(t) = Ce^{kt}$, where:

• $C$ is the initial value ($P(0) = C$).

• $k$ is the growth ($k > 0$) or decay ($k < 0$) rate.

• Applications: Population growth, radioactive decay, compound interest, and more.

Continuous Compound Interest

• For an initial deposit $C$, annual interest rate $r$, compounded $n$ times per year for $t$ years: $P = C(1 + \frac{r}{n})^{nt}$

• As $n \to \infty$, $P = Ce^{rt}$ (continuous compounding).

Logarithmic Functions

Definition and Properties

The logarithmic function with base $b$ is defined as $L = \log_b(K)$ if and only if $b^L = K$, where $b > 0$, $b \neq 1$, and $K > 0$.

• Interpretation: The logarithm answers the question: "To what power must $b$ be raised to obtain $K$?"

• Example: $\log_2(8) = 3$ because $2^3 = 8$.

Domain and Range

• Domain of $\log_b(x)$: $(0, \infty)$ (only positive real numbers).

• Range: $(-\infty, \infty)$ (all real numbers).

Inverse Relationship

• The exponential and logarithmic functions are inverses: $y = b^x$ and $x = \log_b(y)$.

• The graph of $y = \log_b(x)$ is the reflection of $y = b^x$ across the line $y = x$.

Properties of Exponential and Logarithmic Functions

Key Properties and Laws

Exponential Property	Logarithmic Property
$b^0 = 1$	$\log_b(1) = 0$
$b^1 = b$	$\log_b(b) = 1$
$b^{x_1} b^{x_2} = b^{x_1 + x_2}$	$\log_b(MN) = \log_b(M) + \log_b(N)$
	$\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
	$\log_b(M^p) = p \log_b(M)$
$\log_b(b^x) = x$	$b^{\log_b(x)} = x$

Additional info: These properties are essential for simplifying expressions and solving equations involving exponentials and logarithms.

The Natural Logarithm

Definition and Properties

The natural logarithm is the logarithm with base $e$, denoted $\ln(x) = \log_e(x)$. It is the inverse of the natural exponential function $e^x$.

• $\ln(e^x) = x$ for all real $x$

• $e^{\ln(x)} = x$ for all $x > 0$

Applications: Doubling Time and Half-Life

Exponential Growth: Doubling Time

For $P(t) = Ce^{kt}$ with $k > 0$, the doubling time is the time $t$ such that $P(t) = 2C$.

• Set $Ce^{kt} = 2C$

• Divide both sides by $C$: $e^{kt} = 2$

• Take the natural logarithm: $kt = \ln(2)$

• Doubling time: $t = \frac{\ln(2)}{k}$

Exponential Decay: Half-Life

For $k < 0$, the half-life is the time $t$ such that $P(t) = \frac{1}{2}C$.

• Set $Ce^{kt} = \frac{1}{2}C$

• Divide both sides by $C$: $e^{kt} = \frac{1}{2}$

• Take the natural logarithm: $kt = \ln(\frac{1}{2}) = -\ln(2)$

• Half-life: $t = \frac{-\ln(2)}{k}$

Additional info: Doubling time and half-life depend only on the rate constant $k$, not on the initial amount $C$.