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:

• 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: , , , years:

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:

• Quotient Rule:

• Power Rule:

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

Example: Expand

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 Take log:

Example: Solve Rewrite:

Applications: Exponential Growth and Decay

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

• Growth: ,

• Decay: ,

• Half-life:

Example: If , , find :

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: (for ); Asymptote:

• Logarithmic: Domain: ; Range: all real numbers; Asymptote:

Example: For , asymptote is

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 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:

• Solution: The point where both equations are satisfied.

Example: ,

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.

Nonlinear Systems of Equations

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

• Example: ,

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

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 as augmented matrix:

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

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