Q1. Solve the equation:

Background

Topic: Exponential and Logarithmic Equations

This question tests your understanding of logarithmic properties and how to solve equations involving natural logarithms.

Key Terms and Formulas

• Natural logarithm: is the logarithm base .

• Logarithm property:

• To solve , recall that so .

Step-by-Step Guidance

1. Combine the two logarithms using the property to write the equation as a single logarithm.

2. Set the argument of the logarithm equal to 1, since implies .

3. Write the resulting equation in terms of and solve for $x$.

4. Check for any restrictions on (e.g., the domain of the logarithm requires and ).

Try solving on your own before revealing the answer!

Q2. Solve the equation:

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to use logarithmic properties and solve equations where logarithms appear on both sides.

Key Terms and Formulas

• Logarithm property:

• If , then (as long as ).

Step-by-Step Guidance

1. Combine the left side using .

2. Set the arguments of the logarithms equal to each other, since implies .

3. Write the resulting equation in terms of and solve for $x$.

4. Check the domain restrictions for (both and ).

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 where the variable is in the exponent, often by using logarithms.

Key Terms and Formulas

• Exponential equation:

• To solve, take the logarithm of both sides:

• Use the property:

Step-by-Step Guidance

1. Take the natural logarithm of both sides of the equation.

2. Use the property to bring the exponent down.

3. Isolate by solving the resulting linear 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 for when it appears in the exponent, using logarithms.

Key Terms and Formulas

• Exponential equation:

• Take logarithms of both sides to bring down the exponent.

Step-by-Step Guidance

1. Take the natural logarithm of both sides.

2. Apply the property to the left side.

3. Isolate and solve the resulting linear 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 when it appears inside a logarithm, using properties of logarithms and exponentials.

Key Terms and Formulas

• Property:

• To solve , exponentiate both sides:

Step-by-Step Guidance

1. Divide both sides by 3 to isolate .

2. Exponentiate both sides to solve for .

Try solving on your own before revealing the answer!

Q6. Solve the equation:

Background

Topic: Exponential Equations

This question tests your understanding of the exponential function and how to solve for when the exponent is a quadratic expression.

Key Terms and Formulas

• Property: if and only if

Step-by-Step Guidance

1. Set the exponent equal to 0, since only when .

2. Write the resulting quadratic equation in standard form.

3. Solve the quadratic equation for .

Try solving on your own before revealing the answer!