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Step-by-Step Guidance for Exponential and Logarithmic Equations (Precalculus)

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Q1a. Solve for $x$: $4^{x-7} \cdot 2^{x+12} = 0$

Background

Topic: Exponential Equations

This question tests your ability to solve exponential equations by expressing terms with the same base and using properties of exponents.

Key Terms and Formulas:

  • Exponential equation: An equation in which variables appear as exponents.

  • Property: $a^m \cdot a^n = a^{m+n}$

  • Zero Product Property: If $A \cdot B = 0$, then $A = 0$ or $B = 0$.

Step-by-Step Guidance

  1. Rewrite $4^{x-7}$ as $(2^2)^{x-7}$, which simplifies to $2^{2(x-7)}$.

  2. Now, the equation becomes $2^{2(x-7)} \cdot 2^{x+12} = 0$.

  3. Combine the exponents using the property $a^m \cdot a^n = a^{m+n}$ to get a single exponential term.

  4. Set the resulting exponential expression equal to zero and consider what values of $x$ (if any) make this true.

Try solving on your own before revealing the answer!

Q1b. Solve for $x$: $2^{x+12} \cdot 2^{-x} = 7$

Background

Topic: Exponential Equations

This question is similar to the previous one and tests your ability to combine exponents and solve for the variable.

Key Terms and Formulas:

  • Property: $a^m \cdot a^n = a^{m+n}$

  • Inverse operations: Using logarithms to solve for exponents.

Step-by-Step Guidance

  1. Combine the exponents: $2^{x+12} \cdot 2^{-x} = 2^{(x+12) + (-x)} = 2^{12}$.

  2. Set $2^{12} = 7$ and consider how to solve for $x$ (does this make sense?).

  3. Alternatively, check if the equation simplifies to a constant and what that implies for $x$.

Try solving on your own before revealing the answer!

Q1c. Solve for $x$: $2\log_{10}(8x+4) + 6 = 10$

Background

Topic: Logarithmic Equations

This question tests your ability to manipulate logarithmic equations, including using properties of logarithms and solving for the variable inside the log.

Key Terms and Formulas:

  • Property: $a\log_b(x) = \log_b(x^a)$

  • Inverse property: $\log_b(b^x) = x$

  • To isolate $x$, you may need to use exponentiation.

Step-by-Step Guidance

  1. Subtract 6 from both sides to isolate the logarithmic term.

  2. Divide both sides by 2 to further isolate $\log_{10}(8x+4)$.

  3. Rewrite the equation in exponential form to solve for $x$.

  4. Solve for $x$ algebraically.

Try solving on your own before revealing the answer!

Q1d. Solve for $x$: $\ln(x+3) - \ln(x) = \ln(74)$

Background

Topic: Logarithmic Equations

This question tests your understanding of logarithm properties, especially the quotient rule, and solving for the variable inside the log.

Key Terms and Formulas:

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

  • Inverse property: $\ln(e^x) = x$

Step-by-Step Guidance

  1. Combine the logarithms on the left using the quotient rule.

  2. Set the resulting logarithm equal to $\ln(74)$.

  3. Exponentiate both sides to remove the natural log and solve for $x$.

Try solving on your own before revealing the answer!

Q1e. Solve for $x$: $\ln(x+3) + \ln(x) = \ln(70)$

Background

Topic: Logarithmic Equations

This question tests your ability to use the product rule for logarithms and solve for the variable inside the log.

Key Terms and Formulas:

  • Property: $\ln(a) + \ln(b) = \ln(ab)$

  • Inverse property: $\ln(e^x) = x$

Step-by-Step Guidance

  1. Combine the logarithms on the left using the product rule.

  2. Set the resulting logarithm equal to $\ln(70)$.

  3. Exponentiate both sides to remove the natural log and solve for $x$.

Try solving on your own before revealing the answer!

Q1f. Solve for $x$: $\log_{10}(x) = \log_{10}(x) + 1$

Background

Topic: Logarithmic Equations

This question tests your understanding of the properties of logarithms and what happens when the same term appears on both sides of the equation.

Key Terms and Formulas:

  • Property: $\log_b(a) + c = \log_b(a) + c$

  • Recall that $\log_b(a) + 1 = \log_b(a) + \log_b(b)$

Step-by-Step Guidance

  1. Subtract $\log_{10}(x)$ from both sides to isolate the constant.

  2. Interpret what the resulting equation means and whether it is possible for any $x$.

Try solving on your own before revealing the answer!

Q2a. Set up an equation: At time $t$, the patient has 8 mg of atorvastatin in their body.

Background

Topic: Exponential Decay

This question asks you to model a real-world situation using an exponential decay equation.

Key Terms and Formulas:

  • Exponential decay formula: $a(t) = a_0 \cdot \left(\frac{1}{2}\right)^{t/h}$, where $h$ is the half-life.

Step-by-Step Guidance

  1. Write the given formula: $a(t) = 20 \cdot \left(\frac{1}{2}\right)^{t/14}$.

  2. Set $a(t) = 8$ to represent the condition in the problem.

  3. Write the equation: $8 = 20 \cdot \left(\frac{1}{2}\right)^{t/14}$.

Try solving on your own before revealing the answer!

Q2b. Solve the equation from part (a) to find when the level drops below 8 mg.

Background

Topic: Solving Exponential Equations

This question tests your ability to solve for the variable in an exponential decay equation using logarithms.

Key Terms and Formulas:

  • To solve for $t$, use logarithms: $a = a_0 \cdot r^{t}$ leads to $t = \frac{\log(a/a_0)}{\log(r)}$.

Step-by-Step Guidance

  1. Start with the equation: $8 = 20 \cdot \left(\frac{1}{2}\right)^{t/14}$.

  2. Divide both sides by 20 to isolate the exponential term.

  3. Take the natural logarithm of both sides to bring down the exponent.

  4. Use properties of logarithms to solve for $t$.

Try solving on your own before revealing the answer!

Q3a. Set up an equation: When will the turkey reach 165℉?

Background

Topic: Exponential Growth/Decay in Temperature

This question asks you to model a real-world temperature change using an exponential equation.

Key Terms and Formulas:

  • Given: $T(t) = 350 - 290e^{-0.16t}$

Step-by-Step Guidance

  1. Set $T(t) = 165$ to represent the safe temperature.

  2. Write the equation: $165 = 350 - 290e^{-0.16t}$.

Try solving on your own before revealing the answer!

Q3b. Rearrange the equation so that $e^{-0.16t}$ is by itself.

Background

Topic: Solving for an Exponential Term

This question tests your ability to isolate the exponential term in an equation.

Key Terms and Formulas:

  • Inverse operations: Add, subtract, and divide to isolate the exponential.

Step-by-Step Guidance

  1. Start with $165 = 350 - 290e^{-0.16t}$.

  2. Subtract 350 from both sides.

  3. Divide both sides by -290 to isolate $e^{-0.16t}$.

Try solving on your own before revealing the answer!

Q3c. Solve the equation for $t$. Plug your answer into a calculator. Does it seem reasonable?

Background

Topic: Solving Exponential Equations with Logarithms

This question tests your ability to use logarithms to solve for the exponent in an exponential equation.

Key Terms and Formulas:

  • Take the natural logarithm of both sides to solve for $t$.

  • Recall: $\ln(e^x) = x$

Step-by-Step Guidance

  1. Take the natural logarithm of both sides of the equation $e^{-0.16t} = \text{(some value)}$.

  2. Use the property $\ln(e^x) = x$ to bring down the exponent.

  3. Solve for $t$ algebraically.

Try solving on your own before revealing the answer!

Q3d. What happens if you try to solve for when the turkey reaches 375℉?

Background

Topic: Interpreting Exponential Models

This question asks you to interpret the physical meaning of the model and its limitations.

Key Terms and Formulas:

  • Maximum temperature in the model: $T(t) = 350 - 290e^{-0.16t}$

Step-by-Step Guidance

  1. Set $T(t) = 375$ and write the equation: $375 = 350 - 290e^{-0.16t}$.

  2. Try to isolate $e^{-0.16t}$ as before and observe what happens.

  3. Interpret the result: Is it possible for the temperature to reach 375℉ according to this model?

Try solving on your own before revealing the answer!

Q4. How long will it take for a $10,000 investment to be worth at least $25,000 if it increases by 4% per year?

Background

Topic: Exponential Growth (Compound Interest)

This question tests your ability to use the exponential growth formula to solve for time.

Key Terms and Formulas:

  • Exponential growth formula: $A = P(1 + r)^t$

  • $A$ = final amount, $P$ = initial amount, $r$ = growth rate (as a decimal), $t$ = time in years

Step-by-Step Guidance

  1. Set up the equation: $25,000 = 10,000(1.04)^t$.

  2. Divide both sides by 10,000 to isolate the exponential term.

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

  4. Use properties of logarithms to bring down the exponent and solve for $t$.

Try solving on your own before revealing the answer!

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