Q1. Solve the equation: ln x + ln(x − 3) = 0

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to manipulate logarithmic expressions and solve equations involving logarithms.

Key Terms and Formulas:

• (Logarithm addition rule)

Step-by-Step Guidance

1. Combine the logarithms using the addition rule: .

2. Set the combined logarithm equal to zero: .

3. Recall that means . Set .

4. Write the resulting quadratic equation and prepare to solve for .

Try solving on your own before revealing the answer!

Q2. Solve the equation: ln x + ln(x − 3) = ln 4

Background

Topic: Logarithmic Equations

This question tests your ability to solve equations involving logarithms and equate logarithmic expressions.

Key Terms and Formulas:

• If , then

Step-by-Step Guidance

1. Combine the logarithms: .

2. Set the combined logarithm equal to : .

3. Since implies , set .

4. Write the quadratic equation and prepare to solve for .

Try solving on your own before revealing the answer!

Q3. Solve the equation:

Background

Topic: Exponential Equations

This question tests your ability to solve equations involving exponents by expressing both sides with the same base or using logarithms.

Key Terms and Formulas:

• Take logarithms of both sides to solve for

Step-by-Step Guidance

1. Rewrite $21 to see if you can express both sides with base $7$.

2. If not, take the natural logarithm of both sides: .

3. Use the property to simplify: .

4. Isolate by rearranging the equation.

Try solving on your own before revealing the answer!

Q4. Solve the equation:

Background

Topic: Exponential Equations

This question tests your ability to solve exponential equations using logarithms.

Key Terms and Formulas:

Step-by-Step Guidance

1. Take the natural logarithm of both sides: .

2. Apply the exponent rule: .

3. Isolate by rearranging the equation.

Try solving on your own before revealing the answer!

Q5. Solve the equation:

Background

Topic: Logarithmic Equations

This question tests your ability to solve for in an equation involving logarithms.

Key Terms and Formulas:

• is the natural logarithm of

• Exponentiate both sides to solve for

Step-by-Step Guidance

1. Divide both sides by 3: .

2. Exponentiate both sides: .

Try solving on your own before revealing the answer!

Q6. Solve the equation:

Background

Topic: Exponential Equations

This question tests your ability to solve equations where the exponent is a quadratic expression.

Key Terms and Formulas:

• if and only if

Step-by-Step Guidance

1. Set the exponent equal to zero: .

2. Write the quadratic equation and prepare to solve for .

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Q7. Find the exact value:

Background

Topic: Logarithmic Properties

This question tests your ability to use logarithm addition rules and evaluate logarithms.

Key Terms and Formulas:

Step-by-Step Guidance

1. Combine the logarithms: .

2. Calculate .

3. Express the result as a logarithm and prepare to evaluate.

Try solving on your own before revealing the answer!

Q8. Find the exact value:

Background

Topic: Exponential and Logarithmic Properties

This question tests your ability to simplify expressions involving exponentials and logarithms.

Key Terms and Formulas:

Step-by-Step Guidance

1. Rewrite as .

2. Use the property to simplify each term.

3. Multiply the results together.

Try solving on your own before revealing the answer!

Q9. Find the exact value: when and

Background

Topic: Logarithmic Properties

This question tests your ability to use logarithm rules for roots and quotients.

Key Terms and Formulas:

Step-by-Step Guidance

1. Rewrite as .

2. Express as and as .

3. Substitute the given values for and .

4. Set up the expression for evaluation.

Try solving on your own before revealing the answer!