Comprehensive Guide to Logarithm Laws
Logarithm laws are essential tools in mathematics that help simplify and manipulate logarithmic expressions. Understanding these rules enables you to work through complex logarithmic equations more efficiently. This guide breaks down the fundamental laws of logarithms, additional properties, and provides examples for each.
Fundamental Laws of Logarithms
- Product Law:
\( \log_a(mn) = \log_a(m) + \log_a(n) \)
- Example: \( \log(2) + \log(3) = \log(6) \)
- Quotient Law:
\( \log_a\left(\frac{m}{n}\right) = \log_a(m) – \log_a(n) \)
- Example: \( \log(10) – \log(2) = \log(5) \)
- Power Law:
\( \log_a(m^k) = k \cdot \log_a(m) \)
- Example: \( \log(9) = \log(3^2) = 2 \log(3) \)
Additional Logarithmic Properties
- Inverse Property:
\( \log_a(a^k) = k \)
- Example: \( \log_2(2^3) = 3 \)
- Zero Rule:
\( \log_a(1) = 0 \)
- Example: \( \log_5(1) = 0 \)
- Identity Rule:
\( \log_a(a) = 1 \)
- Example: \( \log_7(7) = 1 \)
- Reciprocal Rule:
\( \log_a\left(\frac{1}{m}\right) = -\log_a(m) \)
- Example: \( \log\left(\frac{1}{2}\right) = -\log(2) \)
- Change of Base Rule:
\( \log_a(m) = \frac{\log_c(m)}{\log_c(a)} \)
- Example: \( \log_2(5) = \frac{\log_3(5)}{\log_3(2)} \)
Examples and Solutions
Example 1: Simplifying Using Logarithm Laws
- Simplify \( 2\log(3) + \log(4) \):
\( 2\log(3) + \log(4) = \log(3^2) + \log(4) = \log(9) + \log(4) = \log(36) \)
Example 2: Expanding a Logarithmic Expression
- Expand \( \log(50x^3) \):
\( \log(50x^3) = \log(50) + 3\log(x) \)
Example 3: Solving Using Logarithm Laws
- Simplify \( \log(100) – \log(4) \):
\( \log(100) – \log(4) = \log\left(\frac{100}{4}\right) = \log(25) = 2 \)
Conclusion: Understanding and applying logarithm laws simplifies complex logarithmic expressions. By practicing these rules, you can efficiently solve various mathematical problems.
