College Algebra Review: Logarithms and Exponential Functions

Q1. Find the value of log 180. Give an approximation to four decimal places.

Background

Topic: Common Logarithms

This question tests your ability to evaluate logarithms using a calculator and understand the concept of logarithms as exponents.

Key Terms and Formulas:

• log(x): The logarithm base 10 of x.

• To evaluate log(x), use a scientific calculator or the change of base formula if needed.

Step-by-Step Guidance

1. Recognize that log 180 means $\log_{10} 180$.

2. Use a calculator to find the value of $\log_{10} 180$.

3. Round your answer to four decimal places.

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Q2. Find the value of log 614. Give an approximation to four decimal places.

Background

Topic: Common Logarithms

This question is similar to Q1 and tests your ability to evaluate logarithms using a calculator.

Key Terms and Formulas:

• log(x): The logarithm base 10 of x.

Step-by-Step Guidance

1. Identify that log 614 means $\log_{10} 614$.

2. Use a calculator to compute $\log_{10} 614$.

3. Round your result to four decimal places.

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Q3. Find the value of log 250 + log 34. Give an approximation to four decimal places.

Background

Topic: Logarithm Properties

This question tests your understanding of logarithm addition and the product rule.

Key Terms and Formulas:

• Product Rule: $\log_a b + \log_a c = \log_a (b \times c)$

Step-by-Step Guidance

1. Apply the product rule: $\log_{10} 250 + \log_{10} 34 = \log_{10} (250 \times 34)$.

2. Calculate $250 \times 34$.

3. Find $\log_{10}$ of the result using a calculator.

4. Round your answer to four decimal places.

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Q4. Find the value of log 695 - log 267. Give an approximation to four decimal places.

Background

Topic: Logarithm Properties

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

Key Terms and Formulas:

• Quotient Rule: $\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)$

Step-by-Step Guidance

1. Apply the quotient rule: $\log_{10} 695 - \log_{10} 267 = \log_{10} \left(\frac{695}{267}\right)$.

2. Calculate $\frac{695}{267}$.

3. Find $\log_{10}$ of the result using a calculator.

4. Round your answer to four decimal places.

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Q5. Find the value of ln 101. Give an approximation to four decimal places.

Background

Topic: Natural Logarithms

This question tests your ability to evaluate natural logarithms using a calculator.

Key Terms and Formulas:

• ln(x): The natural logarithm, base $e$.

Step-by-Step Guidance

1. Recognize that ln 101 means $\ln 101$.

2. Use a calculator to find $\ln 101$.

3. Round your answer to four decimal places.

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Q6. Find the value of ln 680 + ln 22. Give an approximation to four decimal places.

Background

Topic: Logarithm Properties

This question tests your understanding of the product rule for natural logarithms.

Key Terms and Formulas:

• Product Rule: $\ln a + \ln b = \ln (a \times b)$

Step-by-Step Guidance

1. Apply the product rule: $\ln 680 + \ln 22 = \ln (680 \times 22)$.

2. Calculate $680 \times 22$.

3. Find $\ln$ of the result using a calculator.

4. Round your answer to four decimal places.

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Q7. Find the value of ln 134 - ln 18. Give an approximation to four decimal places.

Background

Topic: Logarithm Properties

This question tests your understanding of the quotient rule for natural logarithms.

Key Terms and Formulas:

• Quotient Rule: $\ln a - \ln b = \ln \left(\frac{a}{b}\right)$

Step-by-Step Guidance

1. Apply the quotient rule: $\ln 134 - \ln 18 = \ln \left(\frac{134}{18}\right)$.

2. Calculate $\frac{134}{18}$.

3. Find $\ln$ of the result using a calculator.

4. Round your answer to four decimal places.

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Q8. Find the value of ln 4e. Give an approximation to four decimal places.

Background

Topic: Natural Logarithms and Properties

This question tests your understanding of how to evaluate natural logarithms and use properties of $e$.

Key Terms and Formulas:

• $\ln(ab) = \ln a + \ln b$

• $\ln e = 1$

Step-by-Step Guidance

1. Rewrite $\ln 4e$ as $\ln 4 + \ln e$.

2. Recall that $\ln e = 1$.

3. Find $\ln 4$ using a calculator.

4. Add the value of $\ln 4$ to 1.

5. Round your answer to four decimal places.

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Q9. Use the change of base rule to find log base 7 of 8. Give an approximation to four decimal places.

Background

Topic: Change of Base Formula

This question tests your ability to use the change of base formula to evaluate logarithms with bases other than 10 or $e$.

Key Terms and Formulas:

• Change of Base Formula: $\log_a b = \frac{\log_c b}{\log_c a}$, where $c$ is any base (commonly 10 or $e$).

Step-by-Step Guidance

1. Set up the change of base formula: $\log_7 8 = \frac{\log 8}{\log 7}$.

2. Use a calculator to find $\log 8$ and $\log 7$.

3. Divide the two values.

4. Round your answer to four decimal places.

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Q10. Use the change of base rule to find log base 7 of 264.9. Give an approximation to four decimal places.

Background

Topic: Change of Base Formula

This question tests your ability to use the change of base formula for logarithms with non-standard bases.

Key Terms and Formulas:

• Change of Base Formula: $\log_a b = \frac{\log b}{\log a}$

Step-by-Step Guidance

1. Set up the formula: $\log_7 264.9 = \frac{\log 264.9}{\log 7}$.

2. Use a calculator to find $\log 264.9$ and $\log 7$.

3. Divide the two values.

4. Round your answer to four decimal places.

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Q11. Use the change of base rule to find log base 7.3 of 54. Give an approximation to four decimal places.

Background

Topic: Change of Base Formula

This question tests your ability to use the change of base formula for logarithms with decimal bases.

Key Terms and Formulas:

• Change of Base Formula: $\log_a b = \frac{\log b}{\log a}$

Step-by-Step Guidance

1. Set up the formula: $\log_{7.3} 54 = \frac{\log 54}{\log 7.3}$.

2. Use a calculator to find $\log 54$ and $\log 7.3$.

3. Divide the two values.

4. Round your answer to four decimal places.

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Q12. Use the change of base rule to find log base 5 of 0.838. Give an approximation to four decimal places.

Background

Topic: Change of Base Formula

This question tests your ability to use the change of base formula for logarithms with bases other than 10 or $e$.

Key Terms and Formulas:

• Change of Base Formula: $\log_a b = \frac{\log b}{\log a}$

Step-by-Step Guidance

1. Set up the formula: $\log_5 0.838 = \frac{\log 0.838}{\log 5}$.

2. Use a calculator to find $\log 0.838$ and $\log 5$.

3. Divide the two values.

4. Round your answer to four decimal places.

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Q13. Solve the equation $3^x = 23$. Round to the nearest thousandth.

Background

Topic: Exponential Equations

This question tests your ability to solve exponential equations using logarithms.

Key Terms and Formulas:

• To solve $a^x = b$, take the logarithm of both sides.

• $x = \frac{\log b}{\log a}$

Step-by-Step Guidance

1. Take the logarithm of both sides: $\log 3^x = \log 23$.

2. Use the power rule: $x \log 3 = \log 23$.

3. Solve for $x$: $x = \frac{\log 23}{\log 3}$.

4. Use a calculator to evaluate the numerator and denominator.

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Q14. Solve the equation $3^{(x-2)} = 17$. Round to the nearest thousandth.

Background

Topic: Exponential Equations

This question tests your ability to solve exponential equations with shifted exponents.

Key Terms and Formulas:

• Take logarithms of both sides and use the power rule.

Step-by-Step Guidance

1. Take the logarithm of both sides: $\log 3^{(x-2)} = \log 17$.

2. Use the power rule: $(x-2) \log 3 = \log 17$.

3. Solve for $x$: $x-2 = \frac{\log 17}{\log 3}$.

4. Add 2 to both sides to solve for $x$.

5. Use a calculator to evaluate the logarithms.

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Q15. Solve the equation $\log(x-3) = 1 - \log x$. Express the solution in exact form.

Background

Topic: Logarithmic Equations

This question tests your ability to solve equations involving logarithms and algebraic manipulation.

Key Terms and Formulas:

• Logarithm properties: $\log a - \log b = \log \left(\frac{a}{b}\right)$

Step-by-Step Guidance

1. Rewrite the equation: $\log(x-3) + \log x = 1$.

2. Combine logs: $\log \left(\frac{x-3}{x}\right) = 1$.

3. Rewrite in exponential form: $\frac{x-3}{x} = 10^1$.

4. Solve for $x$ algebraically.

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Q16. Solve the equation $\ln(-x) + \ln 4 = \ln(3x-9)$. Express the solution in exact form.

Background

Topic: Logarithmic Equations

This question tests your ability to use properties of logarithms to solve for $x$.

Key Terms and Formulas:

• $\ln a + \ln b = \ln(ab)$

Step-by-Step Guidance

1. Combine logs: $\ln(-x) + \ln 4 = \ln(-4x)$.

2. Set equal to $\ln(3x-9)$: $\ln(-4x) = \ln(3x-9)$.

3. Set the arguments equal: $-4x = 3x-9$.

4. Solve for $x$ algebraically.

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Q17. Solve the equation $\ln e^x - \ln e^9 = \ln e^2$. Express the solution in exact form.

Background

Topic: Logarithm Properties and Exponents

This question tests your ability to use properties of logarithms and exponents.

Key Terms and Formulas:

• $\ln e^a = a$

• $\ln a - \ln b = \ln \left(\frac{a}{b}\right)$

Step-by-Step Guidance

1. Apply $\ln e^x = x$, $\ln e^9 = 9$, $\ln e^2 = 2$.

2. Rewrite the equation: $x - 9 = 2$.

3. Solve for $x$.

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Q18. The growth in population of a city can be seen using the formula $p(t) = 5237e^{0.007t}$, where $t$ is the number of years. According to this formula, how many years will it take the population to double its year 0 value? Round to the nearest tenth of a year.

Background

Topic: Exponential Growth

This question tests your ability to solve for time in an exponential growth model.

Key Terms and Formulas:

• Exponential growth formula: $p(t) = p_0 e^{rt}$

• To double: $p(t) = 2p_0$

Step-by-Step Guidance

1. Set up the equation: $5237e^{0.007t} = 2 \times 5237$.

2. Divide both sides by 5237: $e^{0.007t} = 2$.

3. Take the natural logarithm of both sides: $0.007t = \ln 2$.

4. Solve for $t$: $t = \frac{\ln 2}{0.007}$.

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Q19. The growth in population of a city can be seen using the formula $p(t) = 11,718e^{0.007t}$, where $t$ is the number of years. According to this formula, in how many years will the population reach 17,577? Round to the nearest tenth of a year.

Background

Topic: Exponential Growth

This question tests your ability to solve for time in an exponential growth model when the final population is given.

Key Terms and Formulas:

• Exponential growth formula: $p(t) = p_0 e^{rt}$

Step-by-Step Guidance

1. Set up the equation: $11,718e^{0.007t} = 17,577$.

2. Divide both sides by 11,718: $e^{0.007t} = \frac{17,577}{11,718}$.

3. Take the natural logarithm of both sides: $0.007t = \ln \left(\frac{17,577}{11,718}\right)$.

4. Solve for $t$: $t = \frac{\ln \left(\frac{17,577}{11,718}\right)}{0.007}$.

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Q20. A population is increasing according to the exponential function defined by $y = 3e^{0.04x}$, where $y$ is in millions and $x$ is the number of years. Which of the following should be done in order to answer the question "When will the population reach 4 million?"

Background

Topic: Exponential Functions and Word Problems

This question tests your ability to set up an equation to solve for time when a population reaches a certain value.

Key Terms and Formulas:

• Exponential function: $y = 3e^{0.04x}$

Step-by-Step Guidance

1. Set $y = 4$ in the equation: $3e^{0.04x} = 4$.

2. Divide both sides by 3: $e^{0.04x} = \frac{4}{3}$.

3. Take the natural logarithm of both sides: $0.04x = \ln \left(\frac{4}{3}\right)$.

4. Solve for $x$.

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Q21. A population is increasing according to the exponential function defined by $y = 3e^{0.04x}$, where $y$ is in millions and $x$ is the number of years. Which of the following should be done in order to answer the question "How long will it take for the population to quadruple?"

Background

Topic: Exponential Functions and Word Problems

This question tests your ability to set up an equation to solve for time when a population quadruples.

Key Terms and Formulas:

• Initial population: $y_0 = 3$ million

• Quadruple: $y = 4 \times 3 = 12$ million

Step-by-Step Guidance

1. Set $y = 12$ in the equation: $3e^{0.04x} = 12$.

2. Divide both sides by 3: $e^{0.04x} = 4$.

3. Take the natural logarithm of both sides: $0.04x = \ln 4$.

4. Solve for $x$.

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Q22. The number of books in a small library increases according to the function $B = 8800e^{0.04t}$, where $t$ is measured in years. How many books will the library have after 5 years?

Background

Topic: Exponential Growth

This question tests your ability to evaluate an exponential function for a given value of $t$.

Key Terms and Formulas:

• Exponential growth formula: $B = 8800e^{0.04t}$

Step-by-Step Guidance

1. Plug $t = 5$ into the formula: $B = 8800e^{0.04 \times 5}$.

2. Calculate $0.04 \times 5$.

3. Find $e$ raised to that power.

4. Multiply the result by 8800.

5. Round your answer to the nearest whole number.

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Q23. How long will it take a sample of radioactive substance to decay to half of its original amount, if it decays according to the function $A(t) = 250e^{-0.197t}$, where $t$ is the time in years? Round your answer to the nearest hundredth year.

Background

Topic: Exponential Decay and Half-Life

This question tests your ability to solve for time in an exponential decay model.

Key Terms and Formulas:

• Exponential decay formula: $A(t) = A_0 e^{-kt}$

• Half-life: $A(t) = \frac{1}{2}A_0$

Step-by-Step Guidance

1. Set up the equation: $250e^{-0.197t} = \frac{1}{2} \times 250$.

2. Divide both sides by 250: $e^{-0.197t} = 0.5$.

3. Take the natural logarithm of both sides: $-0.197t = \ln 0.5$.

4. Solve for $t$: $t = \frac{\ln 0.5}{-0.197}$.

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Q24. A sample of 700 grams of radioactive substance decays according to the function $A(t) = 700e^{-0.036t}$, where $t$ is the time in years. How much of the substance will be left in the sample after 10 years? Round your answer to the nearest whole gram.

Background

Topic: Exponential Decay

This question tests your ability to evaluate an exponential decay function for a given time.

Key Terms and Formulas:

• Exponential decay formula: $A(t) = A_0 e^{-kt}$

Step-by-Step Guidance

1. Plug $t = 10$ into the formula: $A(10) = 700e^{-0.036 \times 10}$.

2. Calculate $-0.036 \times 10$.

3. Find $e$ raised to that power.

4. Multiply the result by 700.

5. Round your answer to the nearest whole gram.

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Q25. The amount of particulate matter left in solution during a filtering process is given by the equation $P = 800(2)^{-0.8n}$, where $n$ is the number of filtering steps. Find the amounts left for $n = 0$ and $n = 5$. (Round to the nearest whole number.)

Background

Topic: Exponential Decay and Function Evaluation

This question tests your ability to evaluate an exponential function for different values of $n$.

Key Terms and Formulas:

• Exponential function: $P = 800 \times 2^{-0.8n}$

Step-by-Step Guidance

1. For $n = 0$: Plug $n = 0$ into the formula: $P = 800 \times 2^{-0.8 \times 0}$.

2. Calculate $2^{-0.8 \times 0}$.

3. Multiply the result by 800.

4. For $n = 5$: Plug $n = 5$ into the formula: $P = 800 \times 2^{-0.8 \times 5}$.

5. Calculate $2^{-0.8 \times 5}$.

6. Multiply the result by 800.

7. Round both answers to the nearest whole number.

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