Test Questions for Evaluating Common and Natural Logarithms

📚 Quick Study Guide

• 🔢 Common Logarithm: A logarithm with base 10, denoted as $\log_{10}(x)$ or simply $\log(x)$.
• 🌱 Natural Logarithm: A logarithm with base $e$ (Euler's number, approximately 2.71828), denoted as $\log_e(x)$ or $\ln(x)$.
• ➗ Logarithm Properties:
  • 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: $\log_b(m^p) = p \cdot \log_b(m)$
  • Change of Base Formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$
• 💡 Key Values:
  • $\log(1) = 0$ (because $10^0 = 1$)
  • $\log(10) = 1$ (because $10^1 = 10$)
  • $\ln(1) = 0$ (because $e^0 = 1$)
  • $\ln(e) = 1$ (because $e^1 = e$)
• ➕ Using Calculators: Most calculators have $\log$ (base 10) and $\ln$ (base $e$) functions.

🧪 Practice Quiz

1. What is the value of $\log(100)$?
   1. 0
   2. 1
   3. 2
   4. 3
2. What is the value of $\ln(e^3)$?
   1. 0
   2. 1
   3. 2
   4. 3
3. Simplify: $\log(50) - \log(5)$?
   1. 0
   2. 1
   3. 2
   4. 10
4. What is the value of $\ln(1)$?
   1. -1
   2. 0
   3. 1
   4. e
5. If $\log_2(x) = 3$, what is the value of $x$?
   1. 4
   2. 6
   3. 8
   4. 9
6. Evaluate: $\log(0.01)$?
   1. -2
   2. -1
   3. 0
   4. 2
7. Simplify: $2\ln(e) + 3\log(10)$?
   1. 1
   2. 3
   3. 5
   4. 6