Logarithms Complete Guide

Key Topics Covered

• Rewriting logarithms from exponential form to logarithmic form and vice versa.
• Expanding and condensing logarithms.
• Change of base formula.
• Solving equations involving logarithms.
• Graphing logarithmic functions.

Rewriting Logarithms

From Logarithmic to Exponential Form

• Formula: ( \log_b(x) = n ) can be rewritten as ( b^n = x ).
• Example: ( \log_4(64) = 3 ) becomes ( 4^3 = 64 ).
• Common Log: When no base is specified, it's base 10.

From Exponential to Logarithmic Form

• Formula: ( b^n = x ) can be rewritten as ( \log_b(x) = n ).
• Example: ( 3^4 = 81 ) becomes ( \log_3(81) = 4 ).
• Common logs deal with base 10 if not specified.

Evaluating Logarithms

• Find Value: Convert logs into exponential form to evaluate.
• Examples:
  • ( \log_4(16) \rightarrow 4^x = 16 \rightarrow x = 2 ).
  • ( \log_3(\frac{1}{27}) \rightarrow 3^x = \frac{1}{27} \rightarrow x = -3 ).
• Use calculator with the change of base formula if necessary.

Graphing Logarithmic Functions

• Transformations:
  • Horizontal shifts, vertical shifts affect the graph.
  • Reflect over the line ( y = x ) to find the inverse (logarithmic form).
• Domain and Range:
  • Domain generally ( x > 0 ).
  • Range is all real numbers.

Expanding and Condensing Logs

Properties of Logarithms

• Product Property: ( \log_b(xy) = \log_b(x) + \log_b(y) ).
• Quotient Property: ( \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) ).
• Power Property: ( \log_b(x^n) = n \cdot \log_b(x) ).

Examples

• Expanding:
  • ( \log(f \cdot u \cdot n) \rightarrow \log(f) + \log(u) + \log(n) ).
  • ( \log(\frac{wx}{y}) \rightarrow \log(w) + \log(x) - \log(y) ).
• Condensing:
  • ( \log(a) + \log(b) - \log(c) \rightarrow \log(\frac{ab}{c}) ).

Change of Base Formula

• Formula: ( \log_b(c) = \frac{\log_a(c)}{\log_a(b)} ).
• Use common log (base 10) or natural log (base e).

Solving Logarithmic Equations

• Variable Exponent: Convert to log form to solve.
  • Example: ( 2^{4x} = 5 \rightarrow \log_2(5) = 4x ).
• Using Properties: Simplify using log properties before solving.
• Check Solutions: Ensure solutions are valid (no log of negative number).

Practice and Application

• Practice problems included throughout the guide.
• Encourage practicing rewriting, evaluating, and graphing to reinforce understanding.