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: is the base function.

• Transformation: Adding a constant outside the function, as in , shifts the graph vertically.

• Domain: Set of all possible -values.

• Range: Set of all possible -values.

• y-intercept: The value of .

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

Step-by-Step Guidance

1. Start by recalling the graph of . This function is always positive and increases rapidly as increases.

2. Recognize that adding 6 to shifts the entire graph up by 6 units. So, .

3. Determine the domain: Exponential functions are defined for all real numbers, so consider what this means for .

4. Determine the range: Since is always positive, will always be greater than 6. Think about how to express this in interval notation.

5. Find the y-intercept by evaluating . Substitute into the function and simplify.

6. Identify the horizontal asymptote by considering the behavior of as .

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Q2. Analyze the function :

Background

Topic: Exponential Functions and Transformations

This question is similar to the previous one but involves a reflection and a vertical shift. You are asked to determine the domain, range, y-intercept, and horizontal asymptote.

Key Terms and Formulas:

• Reflection: reflects across the y-axis.

• Vertical shift: Adding 3 shifts the graph up by 3 units.

• Other terms as in Q1.

Step-by-Step Guidance

1. Recall the graph of , which decreases as increases.

2. Adding 3 shifts the graph up by 3 units, so .

3. Determine the domain: Exponential functions are defined for all real .

4. Determine the range: Since is always positive, will always be greater than 3.

5. Find the y-intercept by evaluating . Substitute into the function and simplify.

6. Identify the horizontal asymptote by considering the behavior as .

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Q3. Express as a difference of logarithms.

Background

Topic: Logarithmic Properties

This question tests your knowledge 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. Recall the quotient rule for logarithms.

3. Apply the rule to rewrite as a difference of two logarithms.

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

Background

Topic: Logarithmic Expansion

This question tests your ability to expand logarithmic expressions 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. Apply the power rule to bring the exponents in front of the logarithms.

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Q5. Solve for in .

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:

• Logarithmic Form:

• 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 but do not compute the final value yet.

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Q6. Solve for in .

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:

• Natural Logarithm: is the logarithm with base .

• Inverse Property:

Step-by-Step Guidance

1. Take the natural logarithm of both sides: .

2. Use the property to simplify the left side.

3. Solve for by setting .

4. Set up the expression for but do not compute the final value yet.

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Q7. Write as a 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. Recall the relationship: is equivalent to .

3. Write the logarithmic equation using these values.

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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 argument (), and the result ().

2. Recall that is equivalent to .

3. Write the exponential equation using these values.

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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 Form:

• Exponential Form:

Step-by-Step Guidance

1. Rewrite the logarithmic equation in exponential form: .

2. Calculate to find the value of .

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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 Form:

• Exponential Form:

Step-by-Step Guidance

1. Rewrite the logarithmic equation in exponential form: .

2. Recall that is the same as .

3. Set up the equation and solve for by squaring both sides.

Try solving on your own before revealing the answer!