Properties of Logarithms

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: $log2 (5x) = log2 5 + log_2 x$

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: $log9 (\frac{x}{y}) = log9 x - log_9 y$

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: $log7 \sqrt[3]{w} = log7 w^{\frac{1}{3}} = \frac{1}{3} log_7 w$