Test Questions for Logarithm Change of Base and Calculator Use

📚 Quick Study Guide

• 🧮 Change of Base Formula: The change of base formula allows you to evaluate logarithms with any base using a calculator. 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 (usually 10 or $e$ for calculator use).
• 💡 Common Base 10: When using a calculator, convert to base 10. The formula becomes: $\log_b a = \frac{\log a}{\log b}$. Most calculators have a 'log' button which represents $\log_{10}$.
• 🌱 Natural Log (Base *e*): Another common base is *e* (Euler's number, approximately 2.71828). The natural logarithm is denoted as $\ln$. The formula becomes: $\log_b a = \frac{\ln a}{\ln b}$. Calculators typically have an 'ln' button for natural logarithms.
• ✍️ Calculator Steps (Base 10): To evaluate $\log_b a$ using a calculator: 1) Press the 'log' button. 2) Enter 'a'. 3) Close parentheses (if needed). 4) Press the division symbol '/'. 5) Press the 'log' button again. 6) Enter 'b'. 7) Close parentheses (if needed). 8) Press '=' or 'Enter'.
• ✨ Calculator Steps (Base *e*): To evaluate $\log_b a$ using a calculator: 1) Press the 'ln' button. 2) Enter 'a'. 3) Close parentheses (if needed). 4) Press the division symbol '/'. 5) Press the 'ln' button again. 6) Enter 'b'. 7) Close parentheses (if needed). 8) Press '=' or 'Enter'.
• 🔑 Key Reminders: Always double-check your entries in the calculator, especially when dealing with complex expressions. Parentheses are crucial for correct evaluation.

Practice Quiz

1. What is the change of base formula for logarithms?

   1. $\log_b a = \frac{\log_a b}{\log_c b}$
   2. $\log_b a = \frac{\log_c a}{\log_c b}$
   3. $\log_b a = \log_c a - \log_c b$
   4. $\log_b a = \log_c a + \log_c b$

2. Evaluate $\log_4 16$ using the change of base formula with base 10.

   1. 0.602
   2. 2
   3. 1.386
   4. 2.773

3. Evaluate $\log_2 8$ using the change of base formula with the natural logarithm (ln).

   1. 3
   2. 0.693
   3. 2.079
   4. 1.099

4. What is the value of $\log_5 25$?

   1. 5
   2. 2
   3. 125
   4. 0.301

5. Using a calculator and the change of base formula, approximate $\log_7 49$ to the nearest tenth.

   1. 2.0
   2. 1.7
   3. 2.3
   4. 2.7

6. What is the change of base formula primarily used for?

   1. Simplifying logarithmic expressions.
   2. Evaluating logarithms on calculators with bases other than 10 or *e*.
   3. Combining logarithmic terms.
   4. Solving exponential equations.

7. Evaluate $\log_{1/2} 4$ using the change of base formula with base 10.

   1. 2
   2. -2
   3. 0.5
   4. -0.5