Test Questions: Exponential Regression & Logarithmic Applications (Pre-Calculus)

📚 Quick Study Guide

• 📈 Exponential Regression: A method to model data using an exponential function of the form $y = ab^x$, where $a$ is the initial value, and $b$ is the growth/decay factor. We often use calculators or software to find the best-fit values for $a$ and $b$.
• 🔑 Logarithmic Functions: The inverse of exponential functions. If $y = b^x$, then $x = \log_b y$. They are useful for solving equations where the variable is in the exponent.
• 📊 Applications: These concepts are used in modeling population growth, radioactive decay, compound interest, and various real-world phenomena.
• 📝 Properties of Logarithms: Crucial for simplifying expressions and solving equations:
  • ✨ Product Rule: $\log_b(MN) = \log_b M + \log_b N$
  • ➗ Quotient Rule: $\log_b(\frac{M}{N}) = \log_b M - \log_b N$
  • 💡 Power Rule: $\log_b(M^p) = p \log_b M$
  • 🔄 Change of Base: $\log_b a = \frac{\log_c a}{\log_c b}$
• 🧐 Solving Exponential Equations: Use logarithms to isolate the variable. For example, to solve $a^x = c$, take the logarithm of both sides: $x = \frac{\log c}{\log a}$.

🧪 Practice Quiz

1. What is the general form of an exponential regression equation?
   1. $y = mx + b$
   2. $y = ax^2 + bx + c$
   3. $y = ab^x$
   4. $y = a + bx$
2. The population of a city grows exponentially according to the equation $P(t) = 1000e^{0.05t}$, where $t$ is in years. What is the population after 10 years?
   1. 1050
   2. 1500
   3. 1648.72
   4. 10000
3. Solve for $x$: $2^{x} = 8$
   1. 1
   2. 2
   3. 3
   4. 4
4. Simplify the expression: $\log_2 8 + \log_2 4$
   1. \log_2 12
   2. \log_2 32
   3. 5
   4. 6
5. Which of the following is equivalent to $\log(\frac{a}{b})$?
   1. $\log a + \log b$
   2. $\log a - \log b$
   3. $\log b - \log a$
   4. $\frac{\log a}{\log b}$
6. A radioactive substance decays according to the formula $A(t) = A_0e^{-kt}$, where $A_0$ is the initial amount and $t$ is the time. If the half-life is 10 years, what is the value of $k$?
   1. $\frac{\ln 2}{10}$
   2. $10 \ln 2$
   3. $\frac{10}{\ln 2}$
   4. $\ln(\frac{1}{10})$
7. If $\log_b 9 = 2$, what is the value of $b$?
   1. 3
   2. 4.5
   3. 81
   4. 18