Q1. Find the inverse relation

Background

Topic: Inverse Relations

This question is testing your understanding of how to find the inverse of a given relation, which involves switching the roles of the input and output variables.

Key Terms and Formulas:

• Relation: A set of ordered pairs (x, y).

• Inverse Relation: The set of ordered pairs (y, x) for each (x, y) in the original relation.

Step-by-Step Guidance

1. Write out the given relation as a set of ordered pairs: .

2. For each ordered pair, switch the x and y values to get .

3. List the new set of ordered pairs. This is the inverse relation.

Try solving on your own before revealing the answer!

Q2. Find the inverse function

Background

Topic: Inverse Functions

This question is testing your ability to find the inverse of a function algebraically.

Key Terms and Formulas:

• Function: A rule that assigns each input exactly one output.

• Inverse Function: A function that "undoes" the original function. If is the original function, is its inverse.

Step-by-Step Guidance

1. Replace with in the equation.

2. Switch the variables and in the equation.

3. Solve the new equation for to find .

4. Check your work by composing and to see if you get .

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Q3, Q4, Q5. Evaluate the expression (round to 4 decimals if necessary)

Background

Topic: Evaluating Exponential and Logarithmic Expressions

These questions test your ability to compute values of exponential or logarithmic expressions, possibly using a calculator and rounding as needed.

Key Terms and Formulas:

• Exponential Expression: An expression of the form .

• Logarithmic Expression: An expression of the form .

• Change of Base Formula:

Step-by-Step Guidance

1. Identify whether the expression is exponential or logarithmic.

2. If logarithmic and the base is not 10 or , use the change of base formula if needed.

3. Use your calculator to evaluate the expression, rounding to four decimal places as instructed.

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Q6, Q7. Convert from a logarithm to an exponential or from an exponential to a logarithm

Background

Topic: Logarithmic and Exponential Equations

These questions test your understanding of the relationship between logarithms and exponentials and your ability to rewrite one form as the other.

Key Terms and Formulas:

• Logarithmic Form:

• Exponential Form:

Step-by-Step Guidance

1. Identify the given form (logarithmic or exponential).

2. Recall the equivalence: is the same as .

3. Rewrite the expression in the other form, keeping the base, exponent, and result in the correct positions.

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Q8, Q9, Q10. Simplify the expression using logarithmic properties

Background

Topic: Logarithmic Properties

These questions test your ability to use properties of logarithms to simplify expressions.

Key Terms and Formulas:

• Product Property:

• Quotient Property:

• Power Property:

Step-by-Step Guidance

1. Identify which property (product, quotient, or power) applies to the given expression.

2. Apply the property step by step to break down or combine logarithmic terms as needed.

3. Simplify the expression as much as possible, combining like terms if possible.

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Q11–Q16. Solve the equation (give an exact solution or round to four decimal places when necessary)

Background

Topic: Solving Exponential and Logarithmic Equations

These questions test your ability to solve equations involving exponentials and logarithms, possibly requiring algebraic manipulation and use of logarithmic properties.

Key Terms and Formulas:

• Exponential Equation: An equation where the variable is in the exponent, e.g., .

• Logarithmic Equation: An equation involving logarithms, e.g., .

• Inverse Properties: and

Step-by-Step Guidance

1. Isolate the exponential or logarithmic part of the equation if possible.

2. If exponential, take the logarithm of both sides to bring down the exponent.

3. If logarithmic, rewrite in exponential form or use properties to combine terms.

4. Solve for the variable algebraically, and if necessary, use a calculator to approximate the value to four decimal places.

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Q17–Q24. Application problems

Background

Topic: Applications of Exponential and Logarithmic Functions

These questions involve real-world scenarios where exponential or logarithmic models are used, such as population growth, radioactive decay, or financial calculations.

Key Terms and Formulas:

• Exponential Growth/Decay:

• Compound Interest:

• Logarithmic Models: Used to solve for time or rate in exponential equations.

Step-by-Step Guidance

1. Identify the type of application (growth, decay, finance, etc.).

2. Write down the appropriate model or formula for the scenario.

3. Substitute the given values into the formula.

4. Isolate the unknown variable and solve algebraically, using logarithms if necessary.

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Q25–Q30. Match the graph with the equation

Background

Topic: Graphs of Exponential and Logarithmic Functions

These questions test your ability to recognize the graphs of exponential and logarithmic functions and match them to their equations.

Key Terms and Formulas:

• Exponential Function:

• Logarithmic Function:

• Key Features: Intercepts, asymptotes, domain, and range.

Step-by-Step Guidance

1. Identify the general shape of each graph (increasing, decreasing, asymptotes).

2. Compare the features of each equation (base, vertical/horizontal shifts, reflections) to the graphs.

3. Match each equation to the graph that shares its key features.

Try solving on your own before revealing the answer!