Properties of Logarithms

Product Property

• Definition: logb(m \cdot n) = logb(m) + log_b(n). This property states that the logarithm of a product is the sum of the logarithms.
  • Requirement: The logarithms must have the same base.
• Condensing Logarithms:
  • Example 1: log2 7 + log2 4 = log2 (7 \cdot 4) = log2 28
  • Example 2: log{10} 25 + log{10} 4 = log{10} (25 \cdot 4) = log{10} (100) = 2
  • Example 3: log4 2x + log4 4x^2 = log4 (2x \cdot 4x^2) = log4 8x^3
• Expanding Logarithms:
  • Example 4: log (6) = log (3 \cdot 2) = log(3) + log(2)
  • Example 5: log7 45 = log7 (9 \cdot 5) = log7 9 + log7 5
  • Example 6: log2 (5x) = log2 5 + log_2 x

Quotient Property

• Definition: logb(\frac{m}{n}) = logb(m) - log_b(n). This property states that the logarithm of a quotient is the difference of the logarithms.
• Condensing Logarithms:
  • Example 7: log3 24 - log3 8 = log3 (\frac{24}{8}) = log3 3 = 1
  • Example 8: log2 15 - log2 5 = log2 (\frac{15}{5}) = log2 3
  • Example 9: log4 x - log4 9 = log_4 (\frac{x}{9})
• Expanding Logarithms:
  • Example 10: log8 (\frac{4}{5}) = log8 4 - log_8 5
  • Example 11: log3 (\frac{1}{9}) = log3 1 - log_3 9
  • Example 12: log9 (\frac{x}{y}) = log9 x - log_9 y

Power Property

• Definition: logb (m^n) = n \cdot logb (m). This property states that the logarithm of a number raised to a power is the product of the power and the logarithm of the number.
• Examples:
  • Example 13: log4 3^2 = 2 \cdot log4 3
  • Example 14: log2 x^7 = 7 \cdot log2 x
  • Example 15: log2 \sqrt[3]{8} = log2 (8^{\frac{1}{3}}) = \frac{1}{3} log_2 8
• Expanding using the power property
  • Example 16: log7 8^x = x \cdot log7 8
  • Example 17: 3 \cdot log4 x^{-1} = 3 \cdot (-1) log4 x
  • Example 18: log7 \sqrt[3]{w} = log7 w^{\frac{1}{3}} = \frac{1}{3} log_7 w