College Algebra Practice Test 3 – Step-by-Step Study Guidance

Q1. Find the inverse function for the following functions.

• (1) $f(x) = \frac{2x}{3x-1}$

• (2) $f(x) = \frac{3x-5}{2x+1}$

Background

Topic: Inverse Functions (Rational Functions)

This question tests your ability to find the inverse of a rational function by solving for $x$ in terms of $y$ and then interchanging the variables.

Key Terms and Formulas:

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

• To find the inverse, replace $f(x)$ with $y$, solve for $x$ in terms of $y$, then swap $x$ and $y$.

Step-by-Step Guidance

1. Let $y = f(x)$. For example, $y = \frac{2x}{3x-1}$.

2. Multiply both sides by the denominator to clear fractions: $y(3x-1) = 2x$.

3. Expand and collect all terms involving $x$ on one side: $3xy - y = 2x$.

4. Group $x$ terms together: $3xy - 2x = y$.

5. Factor $x$ out: $x(3y - 2) = y$.

Try solving on your own before revealing the answer!

Q2. Find each of the logarithms.

• 1) $\log 100$

• 2) $\log 0.0001$

• 3) $\log_2 32$

• 4) $\log_4 2$

• 5) $\log_4 1$

• 6) $\log_3 \frac{1}{27}$

Background

Topic: Evaluating Logarithms

This question tests your understanding of logarithm properties and your ability to evaluate logarithms without a calculator.

Key Terms and Formulas:

• Logarithm Definition: $\log_b a = c$ means $b^c = a$.

• Common Logarithm: $\log$ means base 10.

• Properties: $\log_b 1 = 0$, $\log_b b = 1$, $\log_b b^k = k$.

Step-by-Step Guidance

1. For $\log 100$, recall $\log 100 = x$ means $10^x = 100$.

2. For $\log 0.0001$, set $10^x = 0.0001$ and solve for $x$.

3. For $\log_2 32$, set $2^x = 32$ and solve for $x$.

4. For $\log_4 2$, set $4^x = 2$ and solve for $x$.

5. For $\log_4 1$, recall that any base to the power 0 is 1.

6. For $\log_3 \frac{1}{27}$, set $3^x = \frac{1}{27}$ and solve for $x$.

Try solving on your own before revealing the answer!

Q3. Convert to logarithmic equation.

• 1) $8 = 2^x$

• 2) $10^{-2} = 0.01$

• 3) $e^x = Q$

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to rewrite exponential equations in logarithmic form.

Key Terms and Formulas:

• Logarithmic Form: $b^x = a \iff \log_b a = x$

• Natural Logarithm: $e^x = a \iff \ln a = x$

Step-by-Step Guidance

1. For $8 = 2^x$, rewrite as $2^x = 8$ and convert to logarithmic form: $x = \log_2 8$.

2. For $10^{-2} = 0.01$, rewrite as $-2 = \log_{10} 0.01$.

3. For $e^x = Q$, use the natural logarithm: $x = \ln Q$.

Try solving on your own before revealing the answer!

Q4. Convert to exponential equation.

• 1) $\log_3 81 = 4$

• 2) $\log_x B = 5$

• 3) $x = \log_{0.2} B$

Background

Topic: Logarithmic and Exponential Equations

This question tests your ability to rewrite logarithmic equations in exponential form.

Key Terms and Formulas:

• Exponential Form: $\log_b a = c \iff b^c = a$

Step-by-Step Guidance

1. For $\log_3 81 = 4$, rewrite as $3^4 = 81$.

2. For $\log_x B = 5$, rewrite as $x^5 = B$.

3. For $x = \log_{0.2} B$, rewrite as $0.2^x = B$.

Try solving on your own before revealing the answer!

Q5. Find each of the logarithms.

• 1) $\ln 1$

• 2) $\log_5 \sqrt[3]{5}$

• 3) $\ln e$

• 4) $\log \sqrt[4]{10}$

• 5) $\log_5 5^3$

• 6) $\log_{25} 5$

Background

Topic: Evaluating Logarithms and Natural Logarithms

This question tests your understanding of logarithm properties, including roots and exponents.

Key Terms and Formulas:

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

• $\log_b b^k = k$

• $\log_b \sqrt[n]{b} = \frac{1}{n}$

• $\log_b 1 = 0$

Step-by-Step Guidance

1. For $\ln 1$, recall $e^x = 1$ when $x = 0$.

2. For $\log_5 \sqrt[3]{5}$, rewrite $\sqrt[3]{5}$ as $5^{1/3}$.

3. For $\ln e$, recall $e^x = e$ when $x = 1$.

4. For $\log \sqrt[4]{10}$, rewrite $\sqrt[4]{10}$ as $10^{1/4}$.

5. For $\log_5 5^3$, use the property $\log_b b^k = k$.

6. For $\log_{25} 5$, rewrite $25$as $5^2$ and use the change of base formula if needed.

Try solving on your own before revealing the answer!

Q6. Express each of the following in terms of sums and differences of logarithms.

• 1) $\log_a \frac{2\sqrt{b^5 a^3}}{c^4}$

• 2) $\log_b \frac{b y^3}{m^2 n^3}$

Background

Topic: Logarithm Properties (Expansion)

This question tests your ability to use logarithm properties to expand expressions into sums and differences.

Key Terms and Formulas:

• $\log_b \frac{M}{N} = \log_b M - \log_b N$

• $\log_b (MN) = \log_b M + \log_b N$

• $\log_b (M^k) = k \log_b M$

• $\log_b \sqrt[n]{M} = \frac{1}{n} \log_b M$

Step-by-Step Guidance

1. For $\log_a \frac{2\sqrt{b^5 a^3}}{c^4}$, first write the numerator as $2 \cdot (b^5)^{1/2} \cdot (a^3)^{1/2}$.

2. Apply the quotient rule: $\log_a \frac{M}{N} = \log_a M - \log_a N$.

3. Expand the numerator using the product rule: $\log_a (ABC) = \log_a A + \log_a B + \log_a C$.

4. Apply the power rule to each term as needed.

Try solving on your own before revealing the answer!

Q7. Express as a single logarithm.

$3 \log_b x + \frac{1}{2} \log_b y - 2 \log_b z$

Background

Topic: Logarithm Properties (Condensing)

This question tests your ability to combine multiple logarithms into a single logarithm using properties of logarithms.

Key Terms and Formulas:

• $k \log_b M = \log_b M^k$

• $\log_b M + \log_b N = \log_b (MN)$

• $\log_b M - \log_b N = \log_b \frac{M}{N}$

Step-by-Step Guidance

1. Apply the power rule to each term: $3 \log_b x = \log_b x^3$, $\frac{1}{2} \log_b y = \log_b y^{1/2}$, $2 \log_b z = \log_b z^2$.

2. Combine the terms using the sum and difference rules.

Try solving on your own before revealing the answer!

Q8. Given $\log_a 2 = A$ and $\log_a 3 = B$, find each of the following, if possible.

• 1) $\log_a 36$

• 2) $\log_a \frac{4}{9}$

• 3) $\log_a 27$

• 4) $\log_a \frac{1}{8}$

• 5) $\log_a 12$

Background

Topic: Logarithm Properties (Using Given Values)

This question tests your ability to use properties of logarithms and given values to express other logarithms in terms of $A$ and $B$.

Key Terms and Formulas:

• $\log_a (MN) = \log_a M + \log_a N$

• $\log_a \left(\frac{M}{N}\right) = \log_a M - \log_a N$

• $\log_a (M^k) = k \log_a M$

Step-by-Step Guidance

1. Express each argument in terms of powers or products of 2 and 3.

2. Apply the properties to write each logarithm in terms of $A$ and $B$.

Try solving on your own before revealing the answer!

Q9. Simplify each of the following.

• 1) $\log_a a^2$

• 2) $\ln e^{-3}$

• 3) $\log 10^{2x}$

• 4) $3 \log_3 \frac{1}{4} x^2$

• 5) $e^{\ln 3k}$

• 6) $10^{\log 4y}$

Background

Topic: Logarithm and Exponential Simplification

This question tests your ability to use properties of logarithms and exponents to simplify expressions.

Key Terms and Formulas:

• $\log_b b^k = k$

• $\ln e^k = k$

• $e^{\ln x} = x$

• $10^{\log x} = x$

Step-by-Step Guidance

1. Apply the relevant property to each expression to simplify step by step.

2. For expressions with products or quotients inside the log, use expansion first if needed.

Try solving on your own before revealing the answer!

Q10. Solve for $x$.

• (1) $\log_{27} 3 = x$

• (2) $\log_{25} 5 = x$

Background

Topic: Solving Logarithmic Equations

This question tests your ability to convert logarithmic equations to exponential form and solve for $x$.

Key Terms and Formulas:

• $\log_b a = x \iff b^x = a$

• Express the base and argument as powers of the same number if possible.

Step-by-Step Guidance

1. Rewrite the logarithmic equation in exponential form.

2. Express both sides with the same base if possible.

3. Set the exponents equal to each other and solve for $x$.

Try solving on your own before revealing the answer!

Q11. Solve for $x$.

$\log_2 (x + 2) + \log_2 (x - 2) = 2$

Background

Topic: Solving Logarithmic Equations

This question tests your ability to combine logarithms and solve the resulting equation.

Key Terms and Formulas:

• $\log_b M + \log_b N = \log_b (MN)$

• Convert the equation to exponential form after combining logs.

Step-by-Step Guidance

1. Combine the logarithms: $\log_2 ((x + 2)(x - 2)) = 2$.

2. Rewrite as $\log_2 (x^2 - 4) = 2$.

3. Convert to exponential form: $x^2 - 4 = 2^2$.

Try solving on your own before revealing the answer!

Q12. Solve for $x$.

• (1) $3^{2x+3} = 9$

• (2) $4^{3x+2} = 5^{2x-3}$

Background

Topic: Solving Exponential Equations

This question tests your ability to solve exponential equations, possibly by taking logarithms or expressing both sides with the same base.

Key Terms and Formulas:

• For $a^{f(x)} = b^{g(x)}$, take logarithms of both sides if bases are different.

• For $a^{f(x)} = a^k$, set exponents equal: $f(x) = k$.

Step-by-Step Guidance

1. For (1), express $9$as $3^2$ so both sides have base $3$.

2. Set exponents equal: $2x + 3 = 2$.

3. For (2), take the natural logarithm of both sides: $\ln 4^{3x+2} = \ln 5^{2x-3}$.

4. Use the power rule: $(3x+2) \ln 4 = (2x-3) \ln 5$.

Try solving on your own before revealing the answer!

Q13. The Houston-Woodlands-Sugar Land metropolitan area is the fifth largest metropolitan area in the United States. In 2012, the population of this area was 6.18 million, and the exponential growth rate was 2.14% per year.

• (a) Find the exponential growth function.

• (b) Find the doubling time.

Background

Topic: Exponential Growth Models

This question tests your ability to write an exponential growth function and calculate doubling time using the growth rate.

Key Terms and Formulas:

• Exponential growth: $P(t) = P_0 e^{rt}$

• Doubling time: $t = \frac{\ln 2}{r}$

• $P_0$ is the initial population, $r$ is the growth rate (as a decimal), $t$ is time.

Step-by-Step Guidance

1. Write the exponential growth function using $P_0 = 6.18$ million and $r = 0.0214$.

2. For doubling time, use the formula $t = \frac{\ln 2}{r}$ and substitute $r = 0.0214$.

Try solving on your own before revealing the answer!

Q14. Solve the given system.

$\begin{cases} 2x + y = 7 \\ x - 5y = 4 \end{cases}$

Background

Topic: Systems of Linear Equations (2 variables)

This question tests your ability to solve a system of two linear equations using substitution or elimination.

Key Terms and Formulas:

• Substitution method: Solve one equation for one variable, substitute into the other.

• Elimination method: Add or subtract equations to eliminate a variable.

Step-by-Step Guidance

1. Solve the second equation for $x$: $x = 4 + 5y$.

2. Substitute this expression for $x$ into the first equation: $2(4 + 5y) + y = 7$.

3. Solve for $y$.

Try solving on your own before revealing the answer!

Q15. Find the zeros of the polynomial function and state the multiplicity of each.

$f(x) = (x^2 - 5x + 6)^2 (x^2 - 4)$

Background

Topic: Polynomial Zeros and Multiplicity

This question tests your ability to find the zeros of a polynomial and determine their multiplicities.

Key Terms and Formulas:

• Zero: Value of $x$ where $f(x) = 0$.

• Multiplicity: The number of times a zero is repeated (the exponent on the factor).

Step-by-Step Guidance

1. Set each factor equal to zero: $x^2 - 5x + 6 = 0$ and $x^2 - 4 = 0$.

2. Solve each quadratic equation for $x$.

3. For $(x^2 - 5x + 6)^2$, each zero has multiplicity 2.

4. For $(x^2 - 4)$, each zero has multiplicity 1.

Try solving on your own before revealing the answer!

Q16. Solve.

$x^2 - 8 > 6x$

Background

Topic: Quadratic Inequalities

This question tests your ability to solve quadratic inequalities and express the solution in interval notation.

Key Terms and Formulas:

• Move all terms to one side to set the inequality to zero.

• Factor the quadratic if possible.

• Test intervals between the zeros to determine where the inequality holds.

Step-by-Step Guidance

1. Rewrite as $x^2 - 6x - 8 > 0$.

2. Factor the quadratic or use the quadratic formula to find zeros.

3. Test intervals between the zeros to see where the expression is positive.

Try solving on your own before revealing the answer!

Q17. Solve the inequality.

$\frac{x-4}{x+3} - \frac{x+2}{x-1} \leq 0$

Background

Topic: Rational Inequalities

This question tests your ability to solve inequalities involving rational expressions.

Key Terms and Formulas:

• Find a common denominator to combine the expressions.

• Set the numerator and denominator equal to zero to find critical points.

• Test intervals between critical points.

Step-by-Step Guidance

1. Find a common denominator: $(x+3)(x-1)$.

2. Combine the numerators: $\frac{(x-4)(x-1) - (x+2)(x+3)}{(x+3)(x-1)} \leq 0$.

3. Simplify the numerator and set up the inequality.

4. Find zeros of the numerator and denominator to determine test intervals.

Try solving on your own before revealing the answer!

Q18. Solve the logarithmic equation.

$\log x + \log (x + 4) = \log 12$

Background

Topic: Solving Logarithmic Equations

This question tests your ability to combine logarithms and solve for $x$.

Key Terms and Formulas:

• $\log M + \log N = \log (MN)$

• Set arguments equal if logs are equal: $\log A = \log B \implies A = B$

Step-by-Step Guidance

1. Combine the logs: $\log (x(x + 4)) = \log 12$.

2. Set the arguments equal: $x(x + 4) = 12$.

3. Expand and rearrange to form a quadratic equation.

Try solving on your own before revealing the answer!

Q19. For the following function, briefly describe how the graph can be obtained from the graph of a basic logarithmic function. Then graph the function.

$f(x) = 5 - \log (x + 2)$

Background

Topic: Transformations of Logarithmic Functions

This question tests your understanding of how to apply transformations (shifts, reflections, vertical/horizontal changes) to the graph of a logarithmic function.

Key Terms and Formulas:

• Basic log function: $y = \log x$

• Horizontal shift: $y = \log(x + h)$ shifts left by $h$ units if $h > 0$.

• Vertical shift: $y = \log x + k$ shifts up by $k$ units.

• Reflection: $y = -\log x$ reflects over the $x$-axis.

Step-by-Step Guidance

1. Identify the basic function: $y = \log x$.

2. Apply the horizontal shift: $x \to x + 2$ shifts the graph left by 2 units.

3. Apply the reflection and vertical shift: $5 - \log(x + 2)$ reflects over the $x$-axis and shifts up by 5 units.

Try sketching the graph and describing the transformations before revealing the answer!

Q20. Solve the system of equations.

$\begin{cases} x + 6y + 3z = 4 \\ 2x + y + 2z = 3 \\ 3x - 2y + z = 0 \end{cases}$

Background

Topic: Systems of Linear Equations (3 variables)

This question tests your ability to solve a system of three equations in three variables using substitution, elimination, or matrix methods.

Key Terms and Formulas:

• Substitution or elimination methods for systems.

• Matrix methods (optional): Write as $AX = B$ and solve for $X$.

Step-by-Step Guidance

1. Label the equations (1), (2), and (3).

2. Use elimination to eliminate one variable (e.g., $z$) from two pairs of equations.

3. Solve the resulting two-variable system for $x$ and $y$.

4. Back-substitute to find $z$.

Try solving on your own before revealing the answer!