Study Notes on Logarithmic Functions

Logarithmic Functions

• The logarithmic function is defined as:
  $y = ext{log}_b(x)$

• In this function, (b) is the base and (x) is the input.

• Common logarithm bases include 10 (common log) and (e) (natural log).

Key Examples

• Example: ( ext{log}_2(8) = 3 )

  • Means (2^3 = 8)

• Example Evaluations:

  1. ( ext{log}_{10}(100) = 2 )

  2. ( ext{log}_3(27) = 3 )

Graphing Logarithmic Functions

• Example: Find the domain of (F(x) = 3 ext{log}_2(2x - 5))

  • Domain is ( x > 2.5 ) (derived from setting the argument of log > 0).

Transformations

• Transformations can affect the domain and overall graph.

• Example: Shifts, stretches, compressions apply to logarithmic functions.

Homework Assignments

• Convert between exponential and logarithmic forms:

  • Example: (a^{b} = c) translates to (b = ext{log}_a(c))

• Various evaluations of logarithmic expressions, e.g., ( ext{log}_{10}(1,000,000) = 6 ).

• Practice with properties of logarithms such as ( ext{log}b(xy) = ext{log}b(x) + ext{log}_b(y) ).

Common Bases and Their Properties

• Common logarithm is often denoted simply as ( ext{log}(x) ) for base 10.

• Natural logarithm is denoted as ( ext{ln}(x) ) for base (e).

Notable Values

• ( ext{log}_{10}(100) = 2 )

• ( ext{log}_{10}(0.00001) = -5 )

• Ensure operations with logs are conducted within defined domains to avoid undefined expressions.