Precalculus College Algebra Review: Exponential, Logarithmic, and Matrix Problems

Q1. Find the domain of $f(x) = \log(3x - 4)$ in interval notation.

Background

Topic: Logarithmic Functions and Their Domains

This question tests your understanding of the domain of logarithmic functions, specifically when the argument of the logarithm is a linear expression.

Key Terms and Formulas

• Domain: The set of all real numbers for which the function is defined.

• Logarithmic Function: $f(x) = \log_b(x)$ is defined only when $x > 0$.

Step-by-Step Guidance

1. Set the argument of the logarithm greater than zero: $3x - 4 > 0$.

2. Solve the inequality for $x$ to find the values that make the argument positive.

3. Express the solution in interval notation.

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Q2a. How much will $5000 grow into after 3 years if the account earns 7% compounded daily?

Background

Topic: Exponential Growth and Compound Interest

This question tests your ability to use the compound interest formula to determine the future value of an investment with daily compounding.

Key Terms and Formulas

• Compound Interest Formula:

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

• $A$ = future value

• $P$ = principal (initial amount)

• $r$ = annual interest rate (as a decimal)

• $n$ = number of compounding periods per year

• $t$ = number of years

Step-by-Step Guidance

1. Identify the values: $P = 5000$, $r = 0.07$, $n = 365$, $t = 3$.

2. Substitute these values into the compound interest formula.

3. Simplify the expression inside the parentheses and the exponent.

4. Set up the calculation for $A$ but do not compute the final value yet.

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Q2b. Determine the APY for 7% compounded daily.

Background

Topic: Annual Percentage Yield (APY)

This question tests your understanding of how to calculate the effective annual rate when interest is compounded more than once per year.

Key Terms and Formulas

• APY Formula:

$APY = \left(1 + \frac{r}{n}\right)^n - 1$

• $r$ = nominal annual interest rate (as a decimal)

• $n$ = number of compounding periods per year

Step-by-Step Guidance

1. Identify $r = 0.07$ and $n = 365$.

2. Substitute these values into the APY formula.

3. Simplify the expression inside the parentheses and the exponent.

4. Set up the calculation for APY but do not compute the final value yet.

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Q2c. How long will it take for $5000$to grow to $8000$ at 7% compounded daily?

Background

Topic: Solving for Time in Compound Interest Problems

This question tests your ability to solve for the time variable in the compound interest formula, which often involves logarithms.

Key Terms and Formulas

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

• To solve for $t$, you will need to use logarithms.

Step-by-Step Guidance

1. Set $A = 8000$, $P = 5000$, $r = 0.07$, $n = 365$.

2. Write the equation: $8000 = 5000 \left(1 + \frac{0.07}{365}\right)^{365t}$.

3. Divide both sides by $5000$ to isolate the exponential expression.

4. Take the natural logarithm of both sides to solve for $t$.

5. Set up the equation for $t$ but do not solve for the final value yet.

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Q3. Write the expression as a single logarithm: $\frac{1}{2} \log x + \log y$.

Background

Topic: Properties of Logarithms

This question tests your ability to use the properties of logarithms to combine multiple logarithmic terms into a single logarithm.

Key Terms and Formulas

• Power Rule: $a \log_b x = \log_b x^a$

• Product Rule: $\log_b x + \log_b y = \log_b (xy)$

Step-by-Step Guidance

1. Apply the power rule to $\frac{1}{2} \log x$ to rewrite it as a single logarithm.

2. Combine the two logarithms using the product rule.

3. Express the result as a single logarithm.

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Q4a. Find the inverse of $f(x) = 4x + 7$.

Background

Topic: Inverse Functions

This question tests your ability to find the inverse of a linear function.

Key Terms and Formulas

• Inverse Function: If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$.

Step-by-Step Guidance

1. Replace $f(x)$ with $y$ to get $y = 4x + 7$.

2. Swap $x$ and $y$ to get $x = 4y + 7$.

3. Solve for $y$ in terms of $x$ to find the inverse function.

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Q4b. Find the inverse of $g(x) = 4 \log(x - 7)$.

Background

Topic: Inverse Functions Involving Logarithms

This question tests your ability to find the inverse of a function involving a logarithm.

Key Terms and Formulas

• Inverse Function: $f(f^{-1}(x)) = x$

• Exponential and Logarithmic Relationship: $y = \log_b x \iff x = b^y$

Step-by-Step Guidance

1. Replace $g(x)$ with $y$ to get $y = 4 \log(x - 7)$.

2. Swap $x$ and $y$ to get $x = 4 \log(y - 7)$.

3. Solve for $y$ in terms of $x$ by isolating the logarithm and then exponentiating both sides as needed.

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Q5a. Solve for $x$: $2(3^x) + 1 = 29$

Background

Topic: Solving Exponential Equations

This question tests your ability to solve equations where the variable is in the exponent.

Key Terms and Formulas

• Exponential Equation: An equation where the variable appears in the exponent.

• Logarithms: Used to solve for the exponent.

Step-by-Step Guidance

1. Subtract $1$ from both sides to isolate the exponential term.

2. Divide both sides by $2$to further isolate $3^x$.

3. Take the logarithm of both sides to solve for $x$.

4. Set up the equation for $x$ but do not compute the final value yet.

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Q5b. Solve for $x$: $4 \ln(5x) = 12$

Background

Topic: Solving Logarithmic Equations

This question tests your ability to solve equations involving natural logarithms.

Key Terms and Formulas

• Natural Logarithm: $\ln x$ is the logarithm base $e$.

• Exponentiation: Used to undo a logarithm.

Step-by-Step Guidance

1. Divide both sides by $4$to isolate$\ln(5x)$.

2. Exponentiate both sides to remove the logarithm.

3. Solve for $x$ in terms of $e$.

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Q5c. Solve for $x$: $\log_4(x - 2) - \log_4(x - 1) = 1$

Background

Topic: Solving Logarithmic Equations with Properties of Logarithms

This question tests your ability to use properties of logarithms to combine and solve logarithmic equations.

Key Terms and Formulas

• Quotient Rule: $\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)$

• Exponentiation: Used to solve for the variable.

Step-by-Step Guidance

1. Combine the logarithms using the quotient rule.

2. Rewrite the equation in exponential form to solve for $x$.

3. Solve the resulting equation for $x$.

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Q6. Solve the system of equations algebraically: $x^2 - y^2 = 2$, $x^2 + y^2 = 4$.

Background

Topic: Systems of Nonlinear Equations

This question tests your ability to solve a system involving quadratic equations.

Key Terms and Formulas

• System of Equations: Two or more equations with the same variables.

• Substitution or Elimination: Methods for solving systems.

Step-by-Step Guidance

1. Add the two equations to eliminate $y^2$ and solve for $x^2$.

2. Subtract the first equation from the second to eliminate $x^2$ and solve for $y^2$.

3. Take square roots to solve for $x$ and $y$.

4. Check all possible combinations of signs for $x$ and $y$.

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Q7. Solve the system of equations:

$2x + 4y - 5z = 10$ $2x + 4y - 2z = 8$ $-2x - 4y + z = -7$

Write down your RREF matrix and your solution.

Background

Topic: Systems of Linear Equations and Row Reduction

This question tests your ability to solve a system of three equations using matrices and row reduction to reduced row echelon form (RREF).

Key Terms and Formulas

• Augmented Matrix: A matrix representing the system of equations.

• RREF: Reduced Row Echelon Form, used to find solutions.

Step-by-Step Guidance

1. Write the augmented matrix for the system.

2. Use row operations to reduce the matrix to RREF.

3. Interpret the RREF to find the values of $x$, $y$, and $z$.

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Q8. Let $A = \begin{bmatrix} 3 & 2 \\ 0 & 1 \\ 1 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & 1 \\ 5 & 7 \end{bmatrix}$. Determine the following:

Background

Topic: Matrix Operations

This question tests your ability to compute determinants, matrix products, and inverses.

Key Terms and Formulas

• Determinant of a 2x2 Matrix: $\det \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$

• Matrix Multiplication: Multiply rows by columns.

• Inverse of a 2x2 Matrix: $\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

Step-by-Step Guidance

1. For (a), compute the determinant of $B$ using the formula above.

2. For (b), multiply $A$ (3x2) by $B$ (2x2) to find $AB$.

3. For (c), use the determinant from (a) to find $B^{-1}$.

4. For (d), multiply $B$ (2x2) by $A$ (3x2) if possible (check dimensions first).

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Q9. Find the partial fraction decomposition of $\frac{2x - 7}{x^2 - 3x - 4}$.

Background

Topic: Partial Fraction Decomposition

This question tests your ability to decompose a rational function into simpler fractions.

Key Terms and Formulas

• Partial Fractions: Expressing a rational function as a sum of simpler rational expressions.

• Factoring: Factor the denominator first.

Step-by-Step Guidance

1. Factor the denominator $x^2 - 3x - 4$.

2. Set up the decomposition: $\frac{2x - 7}{x^2 - 3x - 4} = \frac{A}{x - 4} + \frac{B}{x + 1}$.

3. Multiply both sides by the denominator to clear fractions.

4. Expand and collect like terms to set up a system of equations for $A$ and $B$.

5. Solve for $A$ and $B$ (set up the equations but do not solve yet).

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