Understanding Logarithms

1. Definition of a Logarithm

• Components: Base (B) and Argument (X)
• Pronunciation: Log base B of X
• Function: Determines the exponent needed for the base to produce the argument.
  • Example:
    • Log base 3 of 9: 3^2 = 9, so answer is 2.
    • Log base 2 of 8: 2^3 = 8, so answer is 3.
    • Log base 2 of 1/8: Rewritten as 2^(-3), so answer is -3.

2. Special Logarithms

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

3. Converting Between Exponential and Logarithmic Equations

• Logarithmic to Exponential: y = log base B of X converts to B^y = X.
  • Example:
    • 5 = log base 2 of 32 converts to 2^5 = 32.
• Exponential to Logarithmic: B^m = n converts to m = log base B of n.
  • Example:
    • 4^3 = 64 converts to 3 = log base 4 of 64.

4. Graph of a Log Function

• Example: y = log base 2 of X.
• Graphing: Easier to pick y-values and calculate corresponding X-values.
• Vertical Asymptote: Exists at x = 0.
• Domain: X > 0.

5. Logarithmic Rules

• Argument Rule: Argument of log must be > 0.
• Base Rule: Base of log must be > 0 and not equal to 1.

6. Power Rule of Logarithms

• Rule: log base B of M^n = n * log base B of M.
  • Examples:
    • log base 3 of 9^4 simplifies to 4 * log base 3 of 9 = 8.
    • log base 2 of sqrt(8) simplifies to 1.5 or 3/2.

7. Product and Quotient Rules

• Product Rule: log base B of M + log base B of N = log base B of (M*N).
  • Example: log base 3 of 7 + log base 3 of 5 = log base 3 of 35.
• Quotient Rule: log base B of M - log base B of N = log base B of (M/N).
  • Example: log base 4 of 30 - log base 4 of 6 = log base 4 of 5.*

8. Other Rules and Tricks

• Change of Base Formula: log base B of M = log base A of M / log base A of B.
• B^log base B of M = M: If base and inside match, it equals the argument.
• Log base A of A = 1.
• Log base B of 1 = 0.

9. Solving Exponential and Logarithmic Equations

• Exponential Equations: Convert to logarithmic form for easier solving.
• Logarithmic Equations: Try to get log = log form to solve.

10. Real-World Applications

• Logarithms: Used to analyze very small/large numbers.
  • Example: pH formula, sound levels.
• Logarithmic Scales: Used to represent wide ranges of data on graphs.

11. Derivative of a Logarithmic Function

• Derivative Rule: d/dx [log base B of X] = 1 / (X * ln(B)).
• Chain Rule for Composites: 1 / (F(x) * ln(B)) * F'(x).
  • Example: Derivative of 2 * log(3x^2) is 4 / (x * ln(10)).*

These points should provide a comprehensive understanding of logarithms, from their definition to applications and derivatives.