Business Calculus Unit 2 Review – Step-by-Step Study Guidance

Q1. Find the value of $\ln 179$ (approximate to four decimal places).

Background

Topic: Natural Logarithms

This question tests your ability to evaluate the natural logarithm (base $e$) of a positive number using a calculator or logarithm properties.

Key Terms and Formulas:

• Natural logarithm: $\ln x$ is the logarithm to the base $e$ (where $e \approx 2.71828$).

• Calculator: Use the $\ln$ button to find the value.

Step-by-Step Guidance

1. Recognize that $\ln 179$ asks for the exponent to which $e$ must be raised to get $179$.

2. Enter $179$into your calculator and press the$\ln$ function to obtain the value.

3. Round your answer to four decimal places as requested.

Try solving on your own before revealing the answer!

Q2. Find the value of $\ln 0.987$ (approximate to four decimal places).

Background

Topic: Natural Logarithms

This question tests your ability to evaluate the natural logarithm of a number less than 1, which should result in a negative value.

Key Terms and Formulas:

• Natural logarithm: $\ln x$

• For $0 < x < 1$, $\ln x$ is negative.

Step-by-Step Guidance

1. Recognize that $\ln 0.987$ will be negative since $0.987 < 1$.

2. Use your calculator: enter $0.987$ and press the $\ln$ button.

3. Round your result to four decimal places.

Try solving on your own before revealing the answer!

Q3. Find the value of $\ln 65,800,000$ (approximate to four decimal places).

Background

Topic: Natural Logarithms

This question tests your ability to compute the natural logarithm of a large number.

Key Terms and Formulas:

• Natural logarithm: $\ln x$

• For large $x$, $\ln x$ will be a large positive number.

Step-by-Step Guidance

1. Enter $65,800,000$ into your calculator.

2. Press the $\ln$ button to compute the value.

3. Round your answer to four decimal places.

Try solving on your own before revealing the answer!

Q4. Solve the exponential equation $e^t = 64$ for $t$. Round your answer to three decimal places if necessary.

Background

Topic: Solving Exponential Equations

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

Key Terms and Formulas:

• Exponential equation: $e^t = a$

• To solve for $t$, take the natural logarithm of both sides: $t = \ln a$

Step-by-Step Guidance

1. Start with the equation: $e^t = 64$.

2. Take the natural logarithm of both sides: $\ln(e^t) = \ln(64)$.

3. Use the property $\ln(e^t) = t$ to simplify: $t = \ln(64)$.

4. Use your calculator to find $\ln(64)$ and round to three decimal places.

Try solving on your own before revealing the answer!

Q5. Solve the exponential equation $e^t = 10,000$ for $t$. Round your answer to three decimal places if necessary.

Background

Topic: Solving Exponential Equations

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

Key Terms and Formulas:

• Exponential equation: $e^t = a$

• To solve for $t$, take the natural logarithm of both sides: $t = \ln a$

Step-by-Step Guidance

1. Start with the equation: $e^t = 10,000$.

2. Take the natural logarithm of both sides: $\ln(e^t) = \ln(10,000)$.

3. Use the property $\ln(e^t) = t$ to simplify: $t = \ln(10,000)$.

4. Use your calculator to find $\ln(10,000)$ and round to three decimal places.

Try solving on your own before revealing the answer!

Q6. Solve the exponential equation $e^{0.03t} = 8$ for $t$. Round your answer to three decimal places if necessary.

Background

Topic: Solving Exponential Equations

This question tests your ability to solve for a variable in the exponent when the base is $e$ and the exponent is a linear expression in $t$.

Key Terms and Formulas:

• Exponential equation: $e^{kt} = a$

• Take the natural logarithm of both sides: $kt = \ln a$

• Solve for $t$: $t = \frac{\ln a}{k}$

Step-by-Step Guidance

1. Start with the equation: $e^{0.03t} = 8$.

2. Take the natural logarithm of both sides: $\ln(e^{0.03t}) = \ln(8)$.

3. Simplify: $0.03t = \ln(8)$.

4. Solve for $t$: $t = \frac{\ln(8)}{0.03}$.

5. Use your calculator to compute $\ln(8)$, then divide by $0.03$ and round to three decimal places.

Try solving on your own before revealing the answer!