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College Algebra Chapter 5 Review: Exponential and Logarithmic Functions

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Q1. Use a calculator to approximate the exponential expression to 6 decimal places:

Background

Topic: Exponential Functions and Euler's Number

This question tests your ability to evaluate exponential expressions using the constant (Euler's number) and to round the result to a specified number of decimal places.

Key Terms and Formulas

  • is approximately 2.718281828...

  • Exponential expression:

Step-by-Step Guidance

  1. Identify the exponent: Here, .

  2. Use a calculator to compute , where .

  3. Make sure your calculator is set to compute exponentials with the base .

  4. After calculating, round your answer to six decimal places.

Try solving on your own before revealing the answer!

Q2. Sketch the graph of the exponential function and complete the table of coordinates.

Background

Topic: Exponential Functions and Graphing

This question tests your understanding of how exponential functions behave and how to plot their graphs using a table of values.

Key Terms and Formulas

  • Exponential function: where

  • For each ,

Step-by-Step Guidance

  1. List the values given: .

  2. For each , calculate .

  3. Fill in the table with the calculated values.

  4. Plot these points on a graph to visualize the exponential curve.

Try solving on your own before revealing the answer!

Q3. Use the graph of and transformations to sketch the exponential function .

Background

Topic: Transformations of Exponential Functions

This question tests your ability to apply vertical shifts to the graph of an exponential function.

Key Terms and Formulas

  • Parent function:

  • Transformation: (vertical shift up by 1 unit)

Step-by-Step Guidance

  1. Identify three points on , such as , , .

  2. Apply the transformation: For each value, add 1 to get the new value.

  3. Write the new coordinates for .

  4. Plot these transformed points to sketch the new graph.

Try solving on your own before revealing the answer!

Q4. Solve the exponential equation by relating the bases:

Background

Topic: Solving Exponential Equations

This question tests your ability to solve exponential equations by expressing both sides with the same base or using logarithms.

Key Terms and Formulas

  • Exponential equation:

  • Properties of exponents:

  • Logarithms:

Step-by-Step Guidance

  1. Rewrite as .

  2. Express and in terms of logarithms if needed.

  3. Take the natural logarithm of both sides to solve for .

  4. Set up the equation for but stop before solving for its value.

Try solving on your own before revealing the answer!

Q5. Use the periodic compound formula to determine the value of given , , , .

Background

Topic: Exponential Growth and Compound Interest

This question tests your ability to use the compound interest formula to calculate the future value of an investment.

Key Terms and Formulas

  • Compound interest formula:

  • = principal, = annual interest rate (as a decimal), = number of compounding periods per year, = number of years

Step-by-Step Guidance

  1. Convert from percent to decimal: .

  2. Plug the values into the formula: .

  3. Calculate and .

  4. Set up the expression for but stop before calculating the final value.

Try solving on your own before revealing the answer!

Q6. Write the logarithmic equation as an exponential equation:

Background

Topic: Logarithmic and Exponential Equations

This question tests your understanding of the relationship between logarithms and exponentials.

Key Terms and Formulas

  • Definition:

  • Here, the base is assumed to be 10 if not specified.

Step-by-Step Guidance

  1. Identify the base of the logarithm (common log, base 10).

  2. Rewrite the equation in exponential form: .

  3. Check that the variables are correctly placed.

Try solving on your own before revealing the answer!

Q7. Evaluate the expression without the use of a calculator:

Background

Topic: Properties of Logarithms

This question tests your ability to use logarithm properties to simplify expressions.

Key Terms and Formulas

Step-by-Step Guidance

  1. Apply the property to each term.

  2. Add the results together.

  3. Write the simplified expression but stop before stating the final value.

Try solving on your own before revealing the answer!

Q8. Use the properties of logarithms to evaluate: for

Background

Topic: Properties of Logarithms

This question tests your understanding of the power rule for logarithms.

Key Terms and Formulas

  • Power rule:

Step-by-Step Guidance

  1. Recognize that uses the power rule.

  2. Apply the rule to simplify the expression.

Try solving on your own before revealing the answer!

Q9. Find the domain of the logarithmic function

Background

Topic: Domain of Logarithmic Functions

This question tests your ability to determine the domain of a logarithmic function by finding where the argument is positive.

Key Terms and Formulas

  • Domain: Set of values for which

  • Logarithm is defined only for positive arguments.

Step-by-Step Guidance

  1. Set .

  2. Factor the quadratic expression.

  3. Find the critical points and determine the intervals where the expression is positive.

  4. Express the domain in interval notation but stop before stating the final answer.

Try solving on your own before revealing the answer!

Q10. Use the quotient rule to expand the logarithmic expression:

Background

Topic: Properties of Logarithms

This question tests your ability to expand logarithmic expressions using the quotient rule.

Key Terms and Formulas

  • Quotient rule:

Step-by-Step Guidance

  1. Apply the quotient rule to the expression.

  2. Simplify using logarithm properties.

  3. Write the expanded expression but stop before stating the final answer.

Try solving on your own before revealing the answer!

Q11. Use the properties of logarithms to evaluate:

Background

Topic: Properties of Logarithms

This question tests your ability to use the product and quotient rules for logarithms.

Key Terms and Formulas

  • Product rule:

  • Quotient rule:

Step-by-Step Guidance

  1. Rewrite as .

  2. Apply the quotient rule: .

  3. Simplify the expression but stop before stating the final value.

Try solving on your own before revealing the answer!

Q12. Use properties of logarithms to rewrite as a single logarithm:

Background

Topic: Properties of Logarithms

This question tests your ability to combine logarithmic expressions using the product rule.

Key Terms and Formulas

  • Product rule:

Step-by-Step Guidance

  1. Apply the product rule to combine the two logarithms.

  2. Simplify the argument of the logarithm.

  3. Write the expression as a single logarithm but stop before stating the final answer.

Try solving on your own before revealing the answer!

Q13. Use the properties of logarithms and the logarithm property of equality to solve:

Background

Topic: Solving Logarithmic Equations

This question tests your ability to solve logarithmic equations by equating arguments or using the change of base formula.

Key Terms and Formulas

  • Logarithm property of equality: If , then

  • Change of base formula:

Step-by-Step Guidance

  1. Rewrite using the change of base formula.

  2. Set equal to the value found from the right side.

  3. Set up the equation for but stop before stating the final value.

Try solving on your own before revealing the answer!

Q14. Use the change of base formula and a calculator to approximate

Background

Topic: Change of Base Formula for Logarithms

This question tests your ability to use the change of base formula and a calculator to approximate logarithmic values.

Key Terms and Formulas

  • Change of base formula:

Step-by-Step Guidance

  1. Apply the change of base formula: .

  2. Use your calculator to compute and .

  3. Divide the results and round to four decimal places.

Try solving on your own before revealing the answer!

Q15. Complete parts a) and b) for the exponential equation

Background

Topic: Solving Exponential Equations Using Logarithms

This question tests your ability to solve exponential equations using natural logarithms and to approximate the solution.

Key Terms and Formulas

  • Natural logarithm:

  • Inverse property:

Step-by-Step Guidance

  1. Take the natural logarithm of both sides: .

  2. Simplify to .

  3. Set up the exact solution: .

  4. Use a calculator to approximate and round to four decimal places.

Try solving on your own before revealing the answer!

Q16. Solve the logarithmic equation:

Background

Topic: Solving Logarithmic Equations

This question tests your ability to use logarithm properties to combine and solve equations.

Key Terms and Formulas

  • Quotient rule:

  • Exponent property:

Step-by-Step Guidance

  1. Combine the logarithms using the quotient rule: .

  2. Rewrite the equation in exponential form: .

  3. Set up the equation for but stop before solving for its value.

Try solving on your own before revealing the answer!

Q17. Use the periodic compound formula to determine the value of given , , , .

Background

Topic: Compound Interest

This question tests your ability to use the compound interest formula to calculate the future value of an investment.

Key Terms and Formulas

  • Compound interest formula:

  • = principal, = annual interest rate (as a decimal), = number of compounding periods per year, = number of years

Step-by-Step Guidance

  1. Convert from percent to decimal: .

  2. Plug the values into the formula: .

  3. Calculate and .

  4. Set up the expression for but stop before calculating the final value.

Try solving on your own before revealing the answer!

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