Understanding Logarithms: Key Concepts

1. What is a Logarithm?

• Components: Logarithmic functions have a base (b) and an argument (x).
• Function: It indicates the exponent to which the base must be raised to get the argument.
• Examples:
  • ( \log_3 9 = 2 ) because ( 3^2 = 9 )
  • ( \log_2 8 = 3 ) because ( 2^3 = 8 )
  • ( \log_2 \frac{1}{8} = -3 ) because ( 2^{-3} = \frac{1}{8} )

2. Special Logarithms

• Common Logarithm: Base 10, written as ( \log x ).
• Natural Logarithm: Base (e) (approx. 2.7), written as ( \ln x ).

3. Converting Between Exponential and Logarithmic Form

• Logarithmic to Exponential: ( y = \log_b x ) becomes ( b^y = x ).
• Exponential to Logarithmic: ( b^m = n ) becomes ( m = \log_b n ).

4. The Graph of a Log Function

• Example: Graph ( y = \log_2 x ).
• Domain: ( x > 0 )
• Properties:
  • Vertical asymptote at ( x = 0 )
  • Domain and base restrictions: Base and argument must be > 0; base cannot be 1.

5. Power Rule of Logarithms

• Rule: ( \log_b (m^n) = n \cdot \log_b m )
• Example Use:
  • ( \log_3 9^4 = 4 \cdot \log_3 9 )

6. Product and Quotient Rules of Logarithms

• Product Rule: ( \log_b (mn) = \log_b m + \log_b n )
• Quotient Rule: ( \log_b \frac{m}{n} = \log_b m - \log_b n )
• Examples:
  • ( \log_3 7 + \log_3 5 = \log_3 (7 \times 5) )
  • ( \log_4 30 - \log_4 6 = \log_4 \frac{30}{6} )

7. Other Rules and Tricks

• Change of Base Formula: ( \log_b m = \frac{\log_a m}{\log_a b} )
• Rule: ( b^{\log_b m} = m )
• Special Values:
  • ( \log_a a = 1 )
  • ( \log_b 1 = 0 )

8. Solving Exponential Equations

• Equation Forms:
  • Convert to logarithmic form when possible.
  • Use the same base when simplifying powers.

9. Solving Logarithmic Equations

• Process:
  • Aim for ( \log = \log ) formats.
  • Check for extraneous solutions that make the argument non-positive.

10. Applications of Logarithms

• Analyzing Numbers: Convert small or large numbers to manageable values.
  • Example: pH = (-\log[H^+])
• Logarithmic Scales: Useful for data with large value spreads.
• Graphing: Logarithmic transformations can make exponential data linear.

11. Derivatives of Logarithmic Functions

• Rule: ( \frac{d}{dx}[\log_b x] = \frac{1}{x \ln b} )
• With Chain Rule: If ( f(x) ) is the argument, multiply by ( f'(x) ).

These concepts provide a foundational understanding of logarithms, their properties, rules for manipulation, and real-world applications.