Exponential and Logarithmic Functions

Properties and Definitions

Exponential and logarithmic functions are fundamental in algebra, modeling growth, decay, and many real-world phenomena. Understanding their properties is essential for solving equations and interpreting graphs.

• Exponential Function: A function of the form , where is the initial value and is the base.

• Logarithmic Function: The inverse of the exponential function, written as .

• Change of Base Formula:

• Product Rule:

• Quotient Rule:

• Power Rule:

Example: because .

Solving Exponential and Logarithmic Equations

To solve equations involving exponentials and logarithms, use properties and inverse operations.

• Exponential Equations: Isolate the exponential term and take logarithms if necessary.

• Logarithmic Equations: Combine logs using properties, then exponentiate both sides to solve for the variable.

Example: Solve by taking logarithms: .

Evaluating Logarithms

Logarithms can often be evaluated without a calculator using properties and known values.

Graphing Exponential and Logarithmic Functions

Graphs of exponential functions show rapid growth or decay, while logarithmic graphs increase slowly and have vertical asymptotes.

• Exponential Graph: passes through and increases (if ) or decreases (if ).

• Logarithmic Graph: has a vertical asymptote at and passes through .

Example: The graph of is increasing and passes through .

Applications of Exponential and Logarithmic Functions

Compound Interest

Exponential functions model compound interest, population growth, and radioactive decay.

• Compound Interest Formula:

• Continuous Compounding:

• Variables: = final amount, = principal, = annual rate, = number of compounding periods per year, = time in years.

Example: If , , , , then .

Radioactive Decay

Radioactive decay is modeled by exponential functions, using half-life to determine the remaining quantity.

• Decay Formula: , where is the half-life.

Example: If grams, years, after $20A = 12\left(\frac{1}{2}\right)^{2} = 3$ grams.

Population Growth

Population growth can be modeled by , where is the initial population and is the growth rate.

• Example: models Austin's population.

Systems of Equations and Matrices

Solving Systems of Equations

Systems of linear equations can be solved using substitution, elimination, or matrix methods.

• Augmented Matrix: Represents a system in matrix form for Gaussian elimination.

• Row-Echelon Form: Used to find solutions systematically.

Example: The system can be written as an augmented matrix and solved.

Matrix Operations

Matrices can be added, subtracted, and multiplied. These operations are useful for solving systems and representing transformations.

• Addition/Subtraction: Element-wise operations.

• Multiplication: Row-by-column multiplication.

Example: Multiply by .

Transformations of Functions

Graphing by Transformations

Functions can be shifted, stretched, or reflected. Understanding transformations helps in sketching graphs and identifying asymptotes.

• Vertical Shift: shifts up/down.

• Horizontal Shift: shifts right/left.

• Reflection: reflects over the x-axis.

Example: is a vertical shift down by 2 units of .

Domain of Functions

Finding the Domain

The domain of a function is the set of all input values for which the function is defined.

• Logarithmic Functions: has domain .

• Exponential Functions: has domain .

Combining and Simplifying Logarithms

Writing as a Single Logarithm

Use properties to combine multiple logarithms into one.

• Example:

Tables: Exponential and Logarithmic Properties

Additional Info

• Questions also cover applications such as investment growth, population modeling, and radioactive decay, which are standard in College Algebra.

• Matrix operations and systems of equations are included, relevant to Ch. 6.

• Transformations and graphing of exponential/logarithmic functions are covered, relevant to Ch. 5 and Ch. 4.