Q1. Solve for in the equation .

Background

Topic: Linear Equations

This question tests your ability to solve a basic linear equation for the variable .

Key Terms and Formulas:

• Linear equation: An equation of the form .

Step-by-Step Guidance

1. Start by getting all terms involving on one side and constants on the other. Subtract from both sides.

2. Next, add $2x$ term.

3. Combine like terms to simplify the equation.

4. Divide both sides by the coefficient of to solve for $x$.

Try solving on your own before revealing the answer!

Q2. Solve for in .

Background

Topic: Logarithmic Equations

This question tests your understanding of logarithms and how to solve for the variable when it appears as the result of a logarithmic expression.

Key Terms and Formulas:

• Logarithm: means .

Step-by-Step Guidance

1. Recognize that means .

2. To solve for , you can use the change of base formula: .

3. Set up the expression .

Try solving on your own before revealing the answer!

Q3. Solve for in .

Background

Topic: Exponential Equations

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

Key Terms and Formulas:

• Exponential equation:

• To solve, take the logarithm of both sides:

Step-by-Step Guidance

1. Take the natural logarithm (or log base 10) of both sides: .

2. Use the property to rewrite the left side.

3. Solve for by dividing both sides by .

Try solving on your own before revealing the answer!

Q4. Solve for in .

Background

Topic: Exponential Equations

This equation involves both a linear and an exponential term. Solving for will require logarithms.

Key Terms and Formulas:

• Exponential equation:

• Logarithms:

Step-by-Step Guidance

1. Isolate the exponential term: .

2. Take the logarithm of both sides: .

3. Use the property to simplify the left side.

4. Set up the equation .

Try solving on your own before revealing the answer!

Q5. Solve for in .

Background

Topic: Logarithmic Equations

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

Key Terms and Formulas:

• Natural logarithm:

Step-by-Step Guidance

1. Rewrite the equation: .

2. Exponentiate both sides to remove the logarithm: .

3. Divide both sides by $2x$.

Try solving on your own before revealing the answer!