Exponential Functions and Their Properties: College Algebra Study Notes

Exponential Functions

Definition and Notation

An exponential function is a mathematical function of the form $f(x) = a^x$, where a is a positive real number not equal to 1, and x is any real number. Exponential functions are fundamental in algebra and have wide applications in science, finance, and engineering.

• Base (a): The constant factor that is raised to a variable exponent.

• Exponent (x): The variable that determines the power to which the base is raised.

• Exponential Growth: Occurs when $a > 1$.

• Exponential Decay: Occurs when $0 < a < 1$.

Properties of Exponents

Exponential expressions follow specific rules, summarized below:

Property	Rule
Quotient Rule	$rac{a^m}{a^n} = a^{m-n}$
Power of a Power	$(a^m)^n = a^{mn}$
Power of a Product	$(ab)^n = a^n b^n$
Power of a Quotient	$( rac{a}{b})^n = rac{a^n}{b^n}$
Zero Exponent	$a^0 = 1$ (for $a eq 0$)
Negative Exponent	$a^{-n} = rac{1}{a^n}$
Fractional Exponent	$a^{1/n} = oot n i a$

Evaluating Exponential Functions

To evaluate an exponential function, substitute the given value of x into the function and compute the result.

• Example 1: If $f(x) = 4^x$, then $f(3) = 4^3 = 64$.

• Example 2: If $f(x) = 5^x$, then $f(-2) = 5^{-2} = rac{1}{25}$.

• Example 3: If $f(x) = ( rac{1}{3})^x$, then $f(-3) = ( rac{1}{3})^{-3} = 3^3 = 27$.

Graphing Exponential Functions

General Characteristics

The graph of $f(x) = a^x$ has distinct features depending on the value of a:

• The points $(-1, rac{1}{a})$, $(0, 1)$, and $(1, a)$ are always on the graph.

• If $a > 1$, the function is increasing.

• If $0 < a < 1$, the function is decreasing.

• The x-axis ($y = 0$) is a horizontal asymptote.

• Domain: $(- , )$ (all real numbers)

• Range: $(0, )$ (all positive real numbers)

Example: Graph of $y = 2^x$

• For $x = -2$: $2^{-2} = rac{1}{4}$

• For $x = -1$: $2^{-1} = rac{1}{2}$

• For $x = 0$: $2^0 = 1$

• For $x = 1$: $2^1 = 2$

• For $x = 2$: $2^2 = 4$

The graph passes through these points and approaches the x-axis but never crosses it.

The Natural Base e

Definition and Properties

The number e is a special mathematical constant approximately equal to 2.71828. It is called the natural base and is used in many applications, especially in calculus and continuous growth models.

• Exponential function with base e: $f(x) = e^x$

• e is a real number, not a variable.

Solving Exponential Equations

One-to-One Property

If $a^u = a^v$, then $u = v$ (for $a > 0$, $a eq 1$). This property allows us to solve equations where both sides have the same base.

Examples

• Example 1: $6^{5x-7} = 216$ $216 = 6^3$ $6^{5x-7} = 6^3$ $5x-7 = 3$ $5x = 10$ $x = 2$

• Example 2: $2^{x+1} = 3^{x-1}$ $2^{x+1} = 3^{x-1}$ (Cannot solve by one-to-one property; requires logarithms. Additional info: This equation would typically be solved using logarithms, which are covered in later sections.)

• Example 3: $( rac{2}{3})^x = rac{125}{8}$ $rac{125}{8} = ( rac{5}{2})^3$ $( rac{2}{3})^x = ( rac{2}{3})^{-3}$ $x = -3$

• Example 4: $e^{4x-1} = e^{-7x}$ $4x-1 = -7x$ $4x + 7x = 1$ $11x = 1$ $x = rac{1}{11}$

Summary Table: Exponent Rules

Rule	Formula
Product of Powers	$a^m a^n = a^{m+n}$
Quotient of Powers	$rac{a^m}{a^n} = a^{m-n}$
Power of a Power	$(a^m)^n = a^{mn}$
Power of a Product	$(ab)^n = a^n b^n$
Power of a Quotient	$( rac{a}{b})^n = rac{a^n}{b^n}$
Zero Exponent	$a^0 = 1$
Negative Exponent	$a^{-n} = rac{1}{a^n}$
Fractional Exponent	$a^{1/n} = oot n i a$

Key Takeaways

• Exponential functions model rapid growth or decay.

• Understanding exponent rules is essential for manipulating exponential expressions.

• Graphs of exponential functions have horizontal asymptotes and never cross the x-axis.

• The natural base e is widely used in mathematics and science.

• Solving exponential equations often involves using the one-to-one property or logarithms.

Additional info: For equations that cannot be solved by matching bases, logarithms are used, which are covered in subsequent sections of College Algebra.