Q1. Analyze the function :

Background

Topic: Exponential Functions and Transformations

This question tests your understanding of how to graph exponential functions, determine their domain and range, find the y-intercept, and identify the horizontal asymptote.

Key Terms and Formulas:

• Exponential function:

• Transformation: Vertical shift by adding 6

• Domain: Set of all possible values

• Range: Set of all possible values

• y-intercept: Value of

• Horizontal asymptote: The line that the function approaches as or

Step-by-Step Guidance

1. Start by recalling the basic graph of and note its properties: domain , range , y-intercept at , and horizontal asymptote at .

2. Apply the vertical shift: Adding 6 to shifts the entire graph up by 6 units. This affects the range, y-intercept, and horizontal asymptote.

3. Determine the new domain and range. The domain remains all real numbers, but the range is shifted up by 6.

4. Find the y-intercept by evaluating .

5. Identify the new horizontal asymptote by considering the behavior as .

Try solving on your own before revealing the answer!

Q2. Analyze the function :

Background

Topic: Exponential Functions and Transformations

This question tests your ability to graph an exponential function with a negative exponent and a vertical shift, and to determine its domain, range, y-intercept, and horizontal asymptote.

Key Terms and Formulas:

• Exponential function:

• Transformation: Vertical shift by adding 3

• Domain: Set of all possible values

• Range: Set of all possible values

• y-intercept: Value of

• Horizontal asymptote: The line that the function approaches as or

Step-by-Step Guidance

1. Recall the graph of : it is a reflection of over the y-axis.

2. Apply the vertical shift: Adding 3 moves the graph up by 3 units.

3. Determine the domain and range after the transformation.

4. Find the y-intercept by evaluating .

5. Identify the horizontal asymptote by considering the end behavior as .

Try solving on your own before revealing the answer!

Q3. Express as a difference of logarithms.

Background

Topic: Logarithmic Properties

This question tests your understanding of the properties of logarithms, specifically how to rewrite the logarithm of a quotient as a difference.

Key Terms and Formulas:

• Quotient Rule for Logarithms:

Step-by-Step Guidance

1. Identify the numerator and denominator inside the logarithm: and .

2. Apply the quotient rule for logarithms to rewrite the expression as a difference.

3. Write the result in terms of and .

Try solving on your own before revealing the answer!

Q4. Rewrite as a sum or difference of multiples of logarithms.

Background

Topic: Logarithmic Expansion

This question tests your ability to expand a logarithmic expression using the product and power rules.

Key Terms and Formulas:

• Product Rule:

• Power Rule:

Step-by-Step Guidance

1. Identify the factors inside the logarithm: , , and .

2. Apply the product rule to separate the logarithm into a sum.

3. Use the power rule to bring the exponents in front of the logarithms.

Try solving on your own before revealing the answer!

Q5. Solve for in the equation .

Background

Topic: Solving Exponential Equations

This question tests your ability to solve for the exponent in an exponential equation using logarithms.

Key Terms and Formulas:

• Exponential equation:

• Logarithms:

• Change of base formula:

Step-by-Step Guidance

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

2. Use the power rule to bring down: .

3. Solve for by dividing both sides by .

4. Set up the expression for in terms of logarithms, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q6. Solve for in the equation .

Background

Topic: Solving Exponential Equations with Base

This question tests your ability to solve for the exponent when the base is using natural logarithms.

Key Terms and Formulas:

• Exponential equation:

• Natural logarithm:

Step-by-Step Guidance

1. Take the natural logarithm of both sides: .

2. Use the property to simplify the left side.

3. Set up the equation , but do not compute the value yet.

Try solving on your own before revealing the answer!

Q7. Write as an equivalent logarithmic equation.

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to convert between exponential and logarithmic forms.

Key Terms and Formulas:

• Exponential form:

• Logarithmic form:

Step-by-Step Guidance

1. Identify the base (), exponent (), and result ().

2. Write the equivalent logarithmic equation using the definition: .

Try solving on your own before revealing the answer!

Q8. Write in equivalent exponential form.

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to convert a logarithmic equation to its equivalent exponential form.

Key Terms and Formulas:

• Logarithmic form:

• Exponential form:

Step-by-Step Guidance

1. Identify the base (), the result (), and the exponent ().

2. Write the equivalent exponential equation: .

Try solving on your own before revealing the answer!

Q9. Determine the value of if .

Background

Topic: Solving Logarithmic Equations

This question tests your ability to solve for the argument of a logarithm given its value.

Key Terms and Formulas:

• Logarithmic equation:

• Equivalent exponential form:

Step-by-Step Guidance

1. Rewrite the logarithmic equation in exponential form: .

2. Set up the calculation for , but do not compute the value yet.

Try solving on your own before revealing the answer!

Q10. Determine the value of if .

Background

Topic: Solving Logarithmic Equations for the Base

This question tests your ability to solve for the base of a logarithm given the argument and the value.

Key Terms and Formulas:

• Logarithmic equation:

• Equivalent exponential form:

Step-by-Step Guidance

1. Rewrite the logarithmic equation in exponential form: .

2. Recall that is the same as , so set up the equation .

3. Set up the next step to solve for by squaring both sides, but do not compute the value yet.

Try solving on your own before revealing the answer!