College Algebra Exam 3 Review Guidance

Q1. Solve $x^2 - 4x = 0$

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations by factoring or using the quadratic formula.

Key Terms and Formulas:

• Quadratic equation: $ax^2 + bx + c = 0$

• Factoring method

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step-by-Step Guidance

1. Rewrite the equation: $x^2 - 4x = 0$

2. Factor out the common term: $x(x - 4) = 0$

3. Set each factor equal to zero: $x = 0$ or $x - 4 = 0$

Try solving on your own before revealing the answer!

Q2. Solve $x^2 - 5x - 8 < 0$

Background

Topic: Quadratic Inequalities

This question tests your ability to solve quadratic inequalities and express the solution as an interval.

Key Terms and Formulas:

• Quadratic inequality

• Finding roots using the quadratic formula

• Test intervals between roots

Step-by-Step Guidance

1. Set the inequality to equality: $x^2 - 5x - 8 = 0$

2. Use the quadratic formula to find the roots: $x = \frac{5 \pm \sqrt{25 + 32}}{2}$

3. Identify the intervals determined by the roots.

4. Test values within each interval to determine where the inequality holds.

Try solving on your own before revealing the answer!

Q3. Solve $x^2 = 0$

Background

Topic: Quadratic Equations

This question tests your ability to solve a simple quadratic equation.

Key Terms and Formulas:

• Quadratic equation

• Factoring

Step-by-Step Guidance

1. Rewrite the equation: $x^2 = 0$

2. Take the square root of both sides: $x = 0$

Try solving on your own before revealing the answer!

Q4. Approximate using a calculator: $2^4$

Background

Topic: Exponents

This question tests your understanding of exponentiation and using a calculator for computation.

Key Terms and Formulas:

• Exponent: $a^n$ means multiplying $a$ by itself $n$ times.

Step-by-Step Guidance

1. Identify the base and exponent: $2^4$

2. Multiply $2$by itself four times: $2 \times 2 \times 2 \times 2$

Try solving on your own before revealing the answer!

Q5. Graph $f(x) = 2^x$ and $g(x) = 2^{-x}$

Background

Topic: Exponential Functions

This question tests your ability to graph exponential functions and understand their behavior.

Key Terms and Formulas:

• Exponential function: $f(x) = a^x$

• Growth vs. decay

Step-by-Step Guidance

1. Identify the base and exponent for each function.

2. For $f(x) = 2^x$, plot points for $x = -2, -1, 0, 1, 2$.

3. For $g(x) = 2^{-x}$, plot points for the same $x$ values.

4. Compare the shapes: $f(x)$ increases, $g(x)$ decreases.

Try graphing on your own before revealing the answer!

Q6. For $f(x) = 3^x$ and $g(x) = 3^{-x}$, graph and compare

Background

Topic: Exponential Functions

This question tests your ability to graph and compare exponential growth and decay.

Key Terms and Formulas:

• Exponential function

• Growth vs. decay

Step-by-Step Guidance

1. Plot $f(x) = 3^x$ for several $x$ values.

2. Plot $g(x) = 3^{-x}$ for the same $x$ values.

3. Compare the graphs: $f(x)$ grows rapidly, $g(x)$ decays.

Try graphing on your own before revealing the answer!

Q7. Invest $10,000. Which yields greater return over 10 years? 12% compounded annually or 12% compounded continuously?

Background

Topic: Exponential Growth and Compound Interest

This question tests your understanding of compound interest formulas and the difference between annual and continuous compounding.

Key Terms and Formulas:

• Annual compounding: $A = P(1 + r)^t$

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

• $P$ = principal, $r$ = rate, $t$ = time

Step-by-Step Guidance

1. Identify the principal ($P = 10,000$), rate ($r = 0.12$), and time ($t = 10$).

2. Set up the formula for annual compounding: $A = 10,000(1 + 0.12)^{10}$

3. Set up the formula for continuous compounding: $A = 10,000e^{0.12 \times 10}$

4. Compare the two expressions to see which yields a greater return.

Try setting up the calculations before revealing the answer!

Q8. Write as an exponential and logarithmic equation: $y = b^x$

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to convert between exponential and logarithmic forms.

Key Terms and Formulas:

• Exponential form: $y = b^x$

• Logarithmic form: $x = \log_b y$

Step-by-Step Guidance

1. Start with the exponential equation: $y = b^x$

2. Rewrite in logarithmic form: $x = \log_b y$

Try converting before revealing the answer!

Q9. Evaluate: $\log_2 64$, $\log_{10} 1000$, $\log_3 27$, $\log_{10} 0.01$, $\ln e^4$

Background

Topic: Logarithms and Natural Logarithms

This question tests your ability to evaluate logarithmic expressions.

Key Terms and Formulas:

• $\log_b a$ is the exponent to which $b$ must be raised to get $a$.

• $\ln a$ is the natural logarithm (base $e$).

Step-by-Step Guidance

1. For each expression, identify the base and the argument.

2. Recall that $\log_b b^n = n$ and $\ln e^n = n$.

3. Apply these properties to each expression.

Try evaluating before revealing the answer!

Q10. Graph: $y = \log_2(x + 2)$, $y = 8 - 2 \log_2 x$

Background

Topic: Logarithmic Functions

This question tests your ability to graph logarithmic functions and understand transformations.

Key Terms and Formulas:

• Logarithmic function: $y = \log_b x$

• Transformations: shifts and stretches

Step-by-Step Guidance

1. Identify the base and argument for each function.

2. For $y = \log_2(x + 2)$, note the horizontal shift.

3. For $y = 8 - 2 \log_2 x$, note the vertical shift and stretch.

4. Plot key points and sketch the general shape.

Try graphing before revealing the answer!

Q11. Simplify: $\ln e^x$, $\ln e^{4x}$

Background

Topic: Properties of Logarithms

This question tests your ability to use logarithm properties to simplify expressions.

Key Terms and Formulas:

• $\ln e^x = x$

• $\ln e^{4x} = 4x$

Step-by-Step Guidance

1. Recall that $\ln e^a = a$ for any exponent $a$.

2. Apply this property to each expression.

Try simplifying before revealing the answer!

Q12. Expand: $\ln \left( \frac{x}{y} \right)$, $\ln \left( x^2 t^3 \right)$

Background

Topic: Logarithm Properties

This question tests your ability to expand logarithmic expressions using properties.

Key Terms and Formulas:

• $\ln \left( \frac{a}{b} \right) = \ln a - \ln b$

• $\ln (ab) = \ln a + \ln b$

• $\ln (a^n) = n \ln a$

Step-by-Step Guidance

1. For $\ln \left( \frac{x}{y} \right)$, use the quotient rule.

2. For $\ln \left( x^2 t^3 \right)$, use the product and power rules.

Try expanding before revealing the answer!

Q13. Condense, write as a single log: $\log_7 x + \log_7 y$

Background

Topic: Logarithm Properties

This question tests your ability to condense logarithmic expressions using properties.

Key Terms and Formulas:

• $\log_b a + \log_b b = \log_b (ab)$

Step-by-Step Guidance

1. Use the product rule to combine the logs.

Try condensing before revealing the answer!

Q14. Use $\log_3 81 = 4$, $\log_5 25 = 2$ to evaluate $\log_3 9$, $\log_5 5$

Background

Topic: Logarithm Properties

This question tests your ability to use known logarithm values to find related values.

Key Terms and Formulas:

• $\log_b b^n = n$

Step-by-Step Guidance

1. Express $9$and $5$ in terms of powers of $3$and $5$ respectively.

2. Apply the property $\log_b b^n = n$.

Try evaluating before revealing the answer!

Q15. Solve: $x = 2^5$, $9^x = 27$, $x^2 = 25$, $x^3 = 27$, $37,500 = 125^x$

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to solve equations involving exponents and logarithms.

Key Terms and Formulas:

• Exponent rules

• Logarithm rules

Step-by-Step Guidance

1. For each equation, rewrite in terms of exponents or logs as needed.

2. Apply the appropriate property to isolate $x$.

Try solving before revealing the answer!

Q16. Solve: $\log_2 x = 3$, $3 \log_{10} x + \log_{10} 15 = 4$

Background

Topic: Logarithmic Equations

This question tests your ability to solve logarithmic equations for $x$.

Key Terms and Formulas:

• Logarithm properties

• Exponentiation to solve for $x$

Step-by-Step Guidance

1. For $\log_2 x = 3$, rewrite in exponential form: $x = 2^3$

2. For $3 \log_{10} x + \log_{10} 15 = 4$, combine logs using properties.

3. Isolate $x$ and solve.

Try solving before revealing the answer!

Q17. Simplify: $\log (2x + 1) + \log 8$

Background

Topic: Logarithm Properties

This question tests your ability to simplify logarithmic expressions using the product rule.

Key Terms and Formulas:

• $\log a + \log b = \log (ab)$

Step-by-Step Guidance

1. Combine the logs using the product rule.

Try simplifying before revealing the answer!