Unit 4 Review: Exponential and Logarithmic Functions, Systems, and Matrices

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 $f(x) = a imes b^x$, where $a$ is the initial value and $b$ is the base.

• Logarithmic Function: The inverse of the exponential function, written as $f(x) = ext{log}_b(x)$.

• Change of Base Formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$

• 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: $\log_b(x^r) = r \log_b(x)$

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

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 $2^x = 5$ by taking logarithms: $x = \log_2(5)$.

Evaluating Logarithms

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

• $\log_{10}(100) = 2$

• $\log_2(8) = 3$

• $\log_5(25) = 2$

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: $f(x) = a b^x$ passes through $(0, a)$ and increases (if $b > 1$) or decreases (if $0 < b < 1$).

• Logarithmic Graph: $f(x) = \log_b(x)$ has a vertical asymptote at $x = 0$ and passes through $(1, 0)$.

Example: The graph of $f(x) = 2^x$ is increasing and passes through $(0, 1)$.

Applications of Exponential and Logarithmic Functions

Compound Interest

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

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

• Continuous Compounding: $A = Pe^{rt}$

• Variables: $A$ = final amount, $P$ = principal, $r$ = annual rate, $n$ = number of compounding periods per year, $t$ = time in years.

Example: If $P = 1000$, $r = 0.05$, $n = 4$, $t = 10$, then $A = 1000\left(1 + \frac{0.05}{4}\right)^{40}$.

Radioactive Decay

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

• Decay Formula: $A = A_0\left(\frac{1}{2}\right)^{t/h}$, where $h$ is the half-life.

Example: If $A_0 = 12$ grams, $h = 10$ years, after $20$years:$A = 12\left(\frac{1}{2}\right)^{2} = 3$ grams.

Population Growth

Population growth can be modeled by $P = P_0 e^{rt}$, where $P_0$ is the initial population and $r$ is the growth rate.

• Example: $P = 660,000 e^{0.017t}$ 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 $\begin{cases} 2x - 2y + 3z = -16 \\ 4x - 3y + 1z = -3 \\ x + y - 5z = -24 \end{cases}$ 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 $\begin{bmatrix}2 & -3 \\ -1 & 4\end{bmatrix}$ by $\begin{bmatrix}-2 & 2 \\ 2 & -2\end{bmatrix}$.

Transformations of Functions

Graphing by Transformations

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

• Vertical Shift: $f(x) + k$ shifts up/down.

• Horizontal Shift: $f(x - h)$ shifts right/left.

• Reflection: $-f(x)$ reflects over the x-axis.

Example: $f(x) = 3^x - 2$ is a vertical shift down by 2 units of $3^x$.

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: $f(x) = \log_a(x - 1)$ has domain $x > 1$.

• Exponential Functions: $f(x) = e^{x+1}$ has domain $(-\infty, \infty)$.

Combining and Simplifying Logarithms

Writing as a Single Logarithm

Use properties to combine multiple logarithms into one.

• Example: $2\ln 3 + \ln 4 - \ln 6 = \ln\left(\frac{3^2 \times 4}{6}\right)$

Tables: Exponential and Logarithmic Properties

Property	Equation	Description
Product Rule	$\log_b(xy) = \log_b(x) + \log_b(y)$	Logarithm of a product equals sum of logarithms
Quotient Rule	$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$	Logarithm of a quotient equals difference of logarithms
Power Rule	$\log_b(x^r) = r \log_b(x)$	Logarithm of a power equals exponent times logarithm
Change of Base	$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$	Converts logarithms to a different base

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.