College Algebra Test 4 Study Guide: Exponential & Logarithmic Functions, Systems, and Matrices

Exponential and Logarithmic Functions

Graphing Exponential Functions Using Transformations

Exponential functions have the form f(x) = a \, b^{x}, where a is a constant and b > 0, b \neq 1. Transformations allow us to shift, stretch, compress, and reflect the graph.

• Vertical Shift: f(x) = a \, b^{x} + k shifts the graph up/down by k.

• Horizontal Shift: f(x) = a \, b^{x-h} shifts the graph right/left by h.

• Reflection: f(x) = -a \, b^{x} reflects the graph over the x-axis.

• Stretch/Compression: Changing a stretches or compresses vertically.

• Asymptote: The horizontal asymptote is y = k for f(x) = a \, b^{x} + k.

Example: Graph f(x) = 2^{x} - 3. The graph of 2^{x} is shifted down by 3 units.

Compound Interest Formulas

Compound interest is calculated using exponential growth formulas.

• Formula: $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• P: Principal (initial investment)

• r: Annual interest rate (decimal)

• n: Number of times interest is compounded per year

• t: Number of years

• A: Amount after t years

Example: $P = 1000$, $r = 0.05$, $n = 4$, $t = 3$ years: $A = 1000 \left(1 + \frac{0.05}{4}\right)^{12}$

Graphing Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and have the form f(x) = \log_{b}(x).

• Domain: x > 0

• Range: All real numbers

• Vertical Asymptote: x = 0

• Transformations: Shifts, stretches, compressions, and reflections similar to exponentials.

Example: Graph f(x) = \log_{2}(x - 1) + 3. Shift right by 1, up by 3.

Properties of Logarithms: Expanding and Condensing

Logarithmic properties allow us to rewrite expressions for simplification.

• Product Rule: $\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)$

• Quotient Rule: $\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)$

• Power Rule: $\log_{b}(x^{r}) = r \log_{b}(x)$

Expanding: Use rules to write as a sum/difference of logs. Condensing: Combine multiple logs into a single log.

Example: Expand $\log_{3}(2x^{4}) = \log_{3}(2) + 4\log_{3}(x)$

Solving Exponential and Logarithmic Equations

Equations involving exponentials or logarithms can be solved using properties and inverse operations.

• Exponential Equations: Isolate the exponential, take logarithms of both sides.

• Logarithmic Equations: Combine logs, rewrite in exponential form.

Example: Solve $2^{x} = 7$ Take log: $x = \log_{2}(7)$

Example: Solve $\log_{3}(x) = 4$ Rewrite: $x = 3^{4} = 81$

Applications: Exponential Growth and Decay

Exponential models describe growth (e.g., populations) and decay (e.g., radioactive substances).

• Growth: $N(t) = N_{0}e^{kt}$, $k > 0$

• Decay: $N(t) = N_{0}e^{-kt}$, $k > 0$

• Half-life: $t_{1/2} = \frac{\ln 2}{k}$

Example: If $N_{0} = 100$, $k = 0.03$, find $N(5)$: $N(5) = 100e^{0.15}$

Domain, Range, and Asymptotes of Exponential & Logarithmic Functions

Understanding domain, range, and asymptotes is essential for graphing and interpreting functions.

• Exponential: Domain: all real numbers; Range: $y > 0$ (for $f(x) = a \, b^{x}$); Asymptote: $y = 0$

• Logarithmic: Domain: $x > 0$; Range: all real numbers; Asymptote: $x = 0$

Example: For $f(x) = 3^{x} + 2$, asymptote is $y = 2$

Systems of Equations and Matrices

Solving Linear Systems by Various Methods

Linear systems can be solved by substitution, elimination, or matrix methods.

• Substitution: Solve one equation for a variable, substitute into the other.

• Elimination: Add/subtract equations to eliminate a variable.

• Matrix Methods: Use augmented matrices and row operations.

Example: Solve $\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}$ by elimination.

Writing and Solving Systems of Linear Equations in Two Variables

Systems of two linear equations can be written in standard form and solved for intersection points.

• Standard Form: $ax + by = c$

• Solution: The point $(x, y)$ where both equations are satisfied.

Example: $3x + 2y = 12$, $x - y = 1$

Applications Involving Systems of Linear Equations

Many real-world problems can be modeled using systems of equations, such as mixture, investment, or motion problems.

• Set up equations based on problem statements.

• Solve for unknowns using appropriate methods.

Example: Two investments total $1000$; one earns 5%, the other 7%. Find amounts if total interest is $60$.

Nonlinear Systems of Equations

Nonlinear systems include at least one equation that is not linear (e.g., quadratic).

• Example: $x^2 + y^2 = 25$, $y = x + 1$

• Solution: Substitute $y$ into the first equation and solve for $x$.

Augmented Matrices and Matrix Solutions

Systems of equations can be represented as augmented matrices for efficient solution using row operations.

• Augmented Matrix: Represents coefficients and constants.

• Row Operations: Used to reach row-echelon form and solve.

• DESMOS Matrix Calculator: Online tool for matrix solutions.

Example: System $\begin{cases} x + 2y = 5 \\ 3x - y = 4 \end{cases}$ as augmented matrix:

x	y	=
1	2	5
3	-1	4

Applications Involving Matrix Solutions

Matrices can be used to solve larger systems and model real-world problems.

• Set up augmented matrix from equations.

• Use row reduction or calculator to find solutions.

Example: Solve a system of three equations using matrix methods.

Summary Table: Key Properties of Exponential and Logarithmic Functions

Function Type	Domain	Range	Asymptote
Exponential	All real numbers	$y > 0$	$y = k$
Logarithmic	$x > 0$	All real numbers	$x = h$

Additional info: The DESMOS matrix calculator is a web-based tool for performing matrix operations and solving systems of equations efficiently.