Exponential and Logarithmic Functions Review – Step-by-Step Study Guidance

Study Guide - Smart Notes

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Q1. The function is given by . Find .

Background

Topic: Inverse Functions of Logarithmic Functions

This question tests your ability to find the inverse of a function involving logarithms. You need to understand how to switch and and solve for the new .

Key Terms and Formulas:

Inverse Function: If , then .
Logarithm Properties: and .

Step-by-Step Guidance

Start by replacing with : .
Switch and to begin finding the inverse: .
Isolate the logarithmic term by subtracting $8x - 8 = 3\log_4(y)$.
Divide both sides by $3\log_4(y)$.

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Q2. What is the solution to the equation ?

Background

Topic: Solving Exponential Equations

This question tests your ability to solve equations where the variable is in the exponent. You may need to isolate the exponential term and use logarithms to solve for .

Key Terms and Formulas:

Exponential Equation: An equation where the variable appears in the exponent.
Logarithm: .

Step-by-Step Guidance

Start by adding $10.
Divide both sides by $4.
Express the result as .
Rewrite $8 and set the exponents equal to each other.

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Q3. What are all of the solutions to the inequality: ?

Background

Topic: Solving Logarithmic Inequalities

This question tests your ability to use properties of logarithms to combine terms and solve inequalities involving logarithmic expressions.

Key Terms and Formulas:

Logarithm Properties: .
Solving Inequalities: If , then (provided ).

Step-by-Step Guidance

Combine the logarithms on the left using the property .
Set up the inequality .
Since the natural log function is increasing, you can drop the logs and write .
Solve the resulting linear inequality for .

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Q4. (a) What’s a logarithmic regression equation that could model the value of Johnny Manziel's rookie card over games played?

Background

Topic: Logarithmic Regression

This question asks you to model real-world data with a logarithmic regression equation, typically of the form or .

Key Terms and Formulas:

Logarithmic Regression: A statistical method to model data that increases or decreases rapidly at first and then levels off.
General Form: or .

Step-by-Step Guidance

Plot the data points (games played vs. card value) to confirm a logarithmic trend.
Use your calculator's regression feature to fit a logarithmic model to the data.
Record the regression equation in the form , where and are constants determined by the regression.

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Q4. (b) How many games would Johnny Manziel need to play for his card to be worth $10?

Background

Topic: Using Regression Equations for Prediction

This question tests your ability to use a regression equation to make predictions about future values.

Key Terms and Formulas:

Regression Equation: (from part a).
Solving for : Set and solve for .

Step-by-Step Guidance

Set the regression equation equal to $10$ (the desired card value).
Isolate the logarithmic term by subtracting from both sides.
Divide by to solve for .
Exponentiate both sides to solve for .

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Q5. Let , , and be positive constants. What is an equivalent expression to ?

Background

Topic: Properties of Logarithms

This question tests your ability to use properties of logarithms to expand or simplify logarithmic expressions.

Key Terms and Formulas:

Quotient Rule:
Product Rule:
Power Rule:

Step-by-Step Guidance

Apply the quotient rule to separate the numerator and denominator.
Apply the product rule to the numerator .
Apply the power rule to each term as needed.
Combine all terms into a simplified expression.

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Q6. The function . The function . Rewrite using properties of logarithms and explain any transformations from to .

Background

Topic: Transformations of Logarithmic Functions

This question tests your understanding of how adding a constant inside the logarithm affects the graph and how to use properties of logarithms to rewrite expressions.

Key Terms and Formulas:

Product Rule:
Power Rule:

Step-by-Step Guidance

Apply the product rule to to separate the terms.
Rewrite using the power rule, since or .
Express in terms of and a constant.
Describe the transformation from to (vertical shift, etc.).

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Q7. Using the tables below, determine if the data given is exponential, logarithmic, or linear. Explain your answer.

Background

Topic: Identifying Function Types from Data

This question tests your ability to recognize patterns in data and match them to linear, exponential, or logarithmic models.

Key Terms and Formulas:

Linear: Constant difference in for constant difference in .
Exponential: Constant ratio in for constant difference in .
Logarithmic: Rapid change at first, then levels off; often, increases by a constant amount as multiplies.

Step-by-Step Guidance

For each table, calculate the differences or ratios between consecutive values.
Compare the patterns to the definitions above to classify each table.
Write a brief explanation for each table based on your calculations.

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Q8. The function . Find the asymptote, domain, range, end behavior, and sketch the graph.

Background

Topic: Graphing Logarithmic Functions

This question tests your understanding of the properties and graphs of logarithmic functions, including transformations.

Key Terms and Formulas:

Vertical Asymptote: Occurs where the argument of the log is zero.
Domain: Set where the argument of the log is positive.
Range: All possible output values.
End Behavior: Behavior as approaches the boundaries of the domain.

Step-by-Step Guidance

Set to find the domain.
Set to find the vertical asymptote.
Recall that the range of a logarithmic function is all real numbers, but check if the transformation affects this.
Analyze the end behavior as approaches the left and right endpoints of the domain.
Sketch the graph based on the above information.

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Q9. (A) The functions and .

Background

Topic: Properties of Logarithms and Exponents

This question tests your ability to combine logarithms and simplify expressions with exponents.

Key Terms and Formulas:

Logarithm Properties: ,
Exponent Properties:

Step-by-Step Guidance

For , use the power rule to rewrite as .
Combine and as a sum of logarithms (cannot combine bases).
For , use the quotient rule for exponents to write as a single power of $2$.
Simplify the exponent by subtracting the denominator exponent from the numerator exponent.

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Q9. (B) The function . Solve for values of in the domain of .

Background

Topic: Solving Logarithmic Equations

This question tests your ability to solve for in equations involving logarithms.

Key Terms and Formulas:

Logarithm Equation:
Solving for : Isolate the log term, then rewrite in exponential form.

Step-by-Step Guidance

Set and write .
Subtract $5$ from both sides to isolate the logarithmic term.
Divide both sides by $6\log_3(x)$.
Rewrite the equation in exponential form to solve for .

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Q9. (C) The function . Find all input values in the domain of that yield an output value of $0$.

Background

Topic: Solving Logarithmic Equations

This question tests your ability to combine logarithmic terms and solve for .

Key Terms and Formulas:

Logarithm Properties: ,
Solving for : Set the expression equal to $0$ and solve.

Step-by-Step Guidance

Set .
Combine the logarithms into a single logarithm using properties.
Set the argument of the logarithm equal to $1\ln(1) = 0$).
Solve the resulting equation for .

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Q10. (A) (i) The function is defined by . Find the value of as a decimal approximation, or indicate that it is not defined. Show the work that leads to your answer.

Background

Topic: Function Composition and Table Lookup

This question tests your ability to evaluate a composite function using a table of values and a given function.

Key Terms and Formulas:

Composite Function:
Table Lookup: Use the table to find , then plug into .

Step-by-Step Guidance

Find from the table.
Plug into to compute .
Simplify the expression for .
Evaluate the result as a decimal (do not round yet).

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Q10. (A) (ii) Find the value of or indicate that it is not defined.

Background

Topic: Inverse Functions and Table Lookup

This question tests your ability to find the input value that produces a given output in a function, using a table.

Key Terms and Formulas:

Inverse Function: is the such that .
Table Lookup: Find where .

Step-by-Step Guidance

Look for in the column of the table.
Identify the corresponding value.
If $40f^{-1}(40)$ is not defined based on the table.

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Q10. (B) (i) Find all values of , as decimal approximations, for which or indicate there are no such values.

Background

Topic: Solving Exponential Equations

This question tests your ability to solve for in an equation involving an exponential function.

Key Terms and Formulas:

Exponential Equation:
Solving for : Isolate the exponential term, then use logarithms to solve for .

Step-by-Step Guidance

Set and write .
Subtract from both sides to isolate .
Divide both sides by to get alone.
Take the logarithm of both sides to solve for .

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Q10. (B) (ii) Determine the end behavior of as decreases without bound. Express your answer using the mathematical notation of a limit.

Background

Topic: End Behavior and Limits of Exponential Functions

This question tests your understanding of how exponential functions behave as approaches negative infinity.

Key Terms and Formulas:

Limit Notation:
Exponential Decay/Growth: approaches $0x \to -\infty$.

Step-by-Step Guidance

Analyze as .
Substitute the behavior into .
Write the limit expression for as .

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Q10. (C) (i) Use the table of values of to determine if is best modeled by a linear, quadratic, exponential, or logarithmic function.

Background

Topic: Identifying Function Types from Data

This question tests your ability to analyze data and determine which type of function best fits the pattern.

Key Terms and Formulas:

Linear: Constant difference in for constant difference in .
Quadratic: Second differences are constant.
Exponential: Constant ratio in for constant difference in .
Logarithmic: increases by a constant amount as multiplies.

Step-by-Step Guidance

Calculate the first differences (change in for each change in ).
Calculate the ratios between consecutive values.
Compare the patterns to the definitions above to decide which model fits best.

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Q10. (C) (ii) Give a reason for your answer in part C (i) based on the relationship between the change in the output values of and the change in the input values of . Refer to the values in the table in your reasoning.

Background

Topic: Justifying Function Type from Data

This question tests your ability to justify your choice of model using evidence from the table.

Key Terms and Formulas:

First Differences: for each .
Ratios: for each step.

Step-by-Step Guidance

Refer to your calculations from part (i).
State whether the differences or ratios are constant, and what that implies about the function type.
Use specific values from the table to support your reasoning.