College Algebra: Logarithms, Exponentials, and Systems of Equations Study Guide

Q1. Expand the logarithmic expression: $\log_3 (7 \cdot 11)$

Background

Topic: Properties of Logarithms (Expanding Logarithmic Expressions)

This question tests your understanding of how to use the product rule for logarithms to expand a single logarithm into a sum of logarithms.

Key Terms and Formulas

• Product Rule: $\log_b (MN) = \log_b M + \log_b N$

Step-by-Step Guidance

1. Identify the base of the logarithm (here, base 3).

2. Recognize that $7 \cdot 11$ is a product inside the logarithm.

3. Apply the product rule to split the logarithm into two separate logarithms with the same base.

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Q2. Expand the logarithmic expression: $\log (100x)$

Background

Topic: Properties of Logarithms (Expanding Logarithmic Expressions)

This question asks you to use the product rule to expand a logarithm with a product inside.

Key Terms and Formulas

• Product Rule: $\log_b (MN) = \log_b M + \log_b N$

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

Step-by-Step Guidance

1. Recognize that $100x$ is a product of $100$and$x$.

2. Apply the product rule to write $\log (100x)$ as $\log 100 + \log x$.

3. Recall that $\log 100$ can be evaluated without a calculator since $100 = 10^2$.

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Q3. Expand the logarithmic expression: $\log_6 \frac{7}{5}$

Background

Topic: Properties of Logarithms (Expanding Logarithmic Expressions)

This question tests your ability to use the quotient rule for logarithms.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Identify the numerator ($7$) and denominator ($5$) inside the logarithm.

2. Apply the quotient rule to separate the logarithm into a difference of two logarithms with the same base.

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Q4. Expand the logarithmic expression: $\log_8 \frac{8}{x}$

Background

Topic: Properties of Logarithms (Expanding Logarithmic Expressions)

This question asks you to use the quotient rule and evaluate logarithms where possible.

Key Terms and Formulas

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

• Logarithm of the base: $\log_b b = 1$

Step-by-Step Guidance

1. Identify the numerator ($8$) and denominator ($x$).

2. Apply the quotient rule to write $\log_8 8 - \log_8 x$.

3. Recall that $\log_8 8 = 1$.

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Q5. Expand the logarithmic expression: $\log_{10} x$

Background

Topic: Properties of Logarithms (Expanding Logarithmic Expressions)

This question checks your understanding of the relationship between logarithms and exponents, and how to express a logarithm in terms of its base.

Key Terms and Formulas

• Common Logarithm: $\log_{10} x$ is often written as $\log x$.

• Logarithm of 10: $\log_{10} 10 = 1$

Step-by-Step Guidance

1. Recognize that $\log_{10} x$ is already in its simplest form, but you can use properties to rewrite it if needed.

2. Consider if you can express $x$ as a product or quotient involving $10$ to use logarithm properties.

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Q6. Expand the logarithmic expression: $\log_b x^5$

Background

Topic: Properties of Logarithms (Expanding Logarithmic Expressions)

This question tests your ability to use the power rule for logarithms.

Key Terms and Formulas

• Power Rule: $\log_b (M^k) = k \log_b M$

Step-by-Step Guidance

1. Identify the exponent ($5$) on$x$ inside the logarithm.

2. Apply the power rule to bring the exponent in front of the logarithm.

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Q7. Expand the logarithmic expression: $\log_b (y z^9)$

Background

Topic: Properties of Logarithms (Expanding Logarithmic Expressions)

This question asks you to use both the product and power rules for logarithms.

Key Terms and Formulas

• Product Rule: $\log_b (MN) = \log_b M + \log_b N$

• Power Rule: $\log_b (M^k) = k \log_b M$

Step-by-Step Guidance

1. Recognize that $y z^9$ is a product of $y$ and $z^9$.

2. Apply the product rule to split into $\log_b y + \log_b z^9$.

3. Apply the power rule to $\log_b z^9$ to bring the exponent in front.

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Q8. Expand the logarithmic expression: $\log_2 \frac{x}{4}$

Background

Topic: Properties of Logarithms (Expanding Logarithmic Expressions)

This question tests your ability to use the quotient rule and the power rule for logarithms.

Key Terms and Formulas

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

• Power Rule: $\log_b (M^k) = k \log_b M$

Step-by-Step Guidance

1. Identify the numerator ($x$) and denominator ($4$).

2. Apply the quotient rule to write $\log_2 x - \log_2 4$.

3. Recall that $4 = 2^2$, so $\log_2 4$ can be simplified using the power rule.

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Q9. Condense the logarithmic expression: $\ln x + 9 \ln y$

Background

Topic: Properties of Logarithms (Condensing Logarithmic Expressions)

This question tests your ability to use the power and product rules in reverse to combine multiple logarithms into a single logarithm.

Key Terms and Formulas

• Power Rule (reverse): $k \log_b M = \log_b M^k$

• Product Rule (reverse): $\log_b M + \log_b N = \log_b (MN)$

Step-by-Step Guidance

1. Use the power rule in reverse to rewrite $9 \ln y$ as $\ln y^9$.

2. Combine $\ln x$ and $\ln y^9$ using the product rule in reverse.

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Q10. Evaluate $\log_{10} 19$ to four decimal places using a calculator.

Background

Topic: Evaluating Logarithms (Calculator Use)

This question tests your ability to use a calculator to evaluate a common logarithm to a specified number of decimal places.

Key Terms and Formulas

• Common Logarithm: $\log_{10} x$ is often written as $\log x$.

Step-by-Step Guidance

1. Enter $19$into your calculator and use the$\log$function to find$\log 19$.

2. Round your answer to four decimal places as requested.

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Q11. Solve the equation: $4^x = 256$ by expressing both sides as powers of the same base.

Background

Topic: Exponential Equations (Equating Exponents)

This question tests your ability to solve exponential equations by rewriting both sides with the same base and then equating exponents.

Key Terms and Formulas

• Exponential Equations: If $a^x = a^y$, then $x = y$.

Step-by-Step Guidance

1. Express $256$as a power of $4$ (find $n$ such that $4^n = 256$).

2. Once both sides have the same base, set the exponents equal to each other.

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Q12. Solve the equation: $2^{5 - 3x} = \frac{1}{16}$ by expressing both sides as powers of the same base.

Background

Topic: Exponential Equations (Equating Exponents)

This question tests your ability to solve exponential equations by expressing both sides with the same base and equating exponents.

Key Terms and Formulas

• Negative Exponents: $a^{-n} = \frac{1}{a^n}$

• Exponential Equations: If $a^x = a^y$, then $x = y$.

Step-by-Step Guidance

1. Express $16$as a power of $2$.

2. Rewrite $\frac{1}{16}$ as $2^{-4}$ using negative exponents.

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

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Q13. Solve the equation: $3^{3x - 6} = 27$ by expressing both sides as powers of the same base.

Background

Topic: Exponential Equations (Equating Exponents)

This question tests your ability to solve exponential equations by expressing both sides with the same base and equating exponents.

Key Terms and Formulas

• Exponential Equations: If $a^x = a^y$, then $x = y$.

Step-by-Step Guidance

1. Express $27$as a power of $3$.

2. Once both sides have the same base, set the exponents equal to each other and solve for $x$.

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Q14. Solve the exponential equation: $4^{x + 7} = 6$ and express the solution in terms of natural logarithms.

Background

Topic: Exponential Equations (Solving Using Logarithms)

This question tests your ability to solve exponential equations by taking logarithms of both sides and isolating the variable.

Key Terms and Formulas

• Logarithms: $a^x = b \implies x = \frac{\ln b}{\ln a}$

• Properties of Logarithms: $\ln a^x = x \ln a$

Step-by-Step Guidance

1. Take the natural logarithm of both sides: $\ln(4^{x+7}) = \ln 6$.

2. Use the power rule to bring the exponent down: $(x+7) \ln 4 = \ln 6$.

3. Solve for $x$ by isolating it on one side of the equation.

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Q15. Solve the exponential equation: $e^{5x} = 2$ and express the solution in terms of natural logarithms.

Background

Topic: Exponential Equations (Solving Using Logarithms)

This question tests your ability to solve for the exponent in an equation involving the natural exponential function.

Key Terms and Formulas

• Natural Logarithm: $\ln e^x = x$

• Solving Exponential Equations: Take the natural logarithm of both sides to isolate the exponent.

Step-by-Step Guidance

1. Take the natural logarithm of both sides: $\ln(e^{5x}) = \ln 2$.

2. Use the property $\ln(e^{5x}) = 5x$.

3. Solve for $x$ by dividing both sides by $5$.

Try solving on your own before revealing the answer!