Test Questions on the Change of Base Formula for Logarithms

📚 Quick Study Guide

• 🔑 The Change of Base Formula: This formula allows you to convert logarithms from one base to another. It's super useful when your calculator can't handle a specific base! The formula is: $\log_b a = \frac{\log_c a}{\log_c b}$, where $a$ is the argument, $b$ is the original base, and $c$ is the new base.
• 💡 Choosing a New Base: Usually, you'll want to choose a new base that your calculator can handle, like base 10 (common logarithm) or base $e$ (natural logarithm). So, you'd rewrite the formula as either $\log_b a = \frac{\log_{10} a}{\log_{10} b}$ or $\log_b a = \frac{\ln a}{\ln b}$.
• ➕ Properties of Logarithms: Remember the basic properties of logarithms, as they can often simplify calculations before applying the change of base formula. These include the product rule, quotient rule, and power rule.
  • ➕ Product Rule: $\log_b(xy) = \log_b x + \log_b y$
  • ➗ Quotient Rule: $\log_b(\frac{x}{y}) = \log_b x - \log_b y$
  • ⚡ Power Rule: $\log_b(x^p) = p \log_b x$
• ✍️ Example: Let's say you want to find $\log_5 20$. Using the change of base formula with base 10, you get $\log_5 20 = \frac{\log_{10} 20}{\log_{10} 5} \approx \frac{1.301}{0.699} \approx 1.86$.

Practice Quiz

1. What is the change of base formula for logarithms?
1. $\log_b a = \log_a b$
2. $\log_b a = \frac{\log_c b}{\log_c a}$
3. $\log_b a = \frac{\log_c a}{\log_c b}$
4. $\log_b a = \log_c a - \log_c b$
2. Evaluate $\log_8 16$ using the change of base formula with base 2.
1. $\frac{3}{4}$
2. $\frac{4}{3}$
3. $\frac{1}{2}$
4. 2
3. Which of the following is equivalent to $\log_3 7$ using the natural logarithm (ln)?
1. $\frac{\log 7}{\log 3}$
2. $\frac{\ln 3}{\ln 7}$
3. $\frac{\ln 7}{\ln 3}$
4. $\ln 7 - \ln 3$
4. Simplify $\log_4 8 + \log_4 2$ using logarithm properties. Then evaluate using change of base if needed.
1. 1
2. 2
3. 3
4. 4
5. Calculate $\log_9 27$
1. $\frac{2}{3}$
2. $\frac{3}{2}$
3. 2
4. 3
6. Rewrite $\log_7 (x+2)$ in terms of the common logarithm (base 10).
1. $\frac{\log (x+2)}{\log 7}$
2. $\frac{\log 7}{\log (x+2)}$
3. $\log 7 - \log (x+2)$
4. $\log (x+2) - \log 7$
7. If $\log_b 12 = 2.485$ and $\log_b 3 = 0.792$, find $\log_b 4$
1. 1.693
2. 3.277
3. 0.243
4. 2