2.12 Logarithmic Function Manipulation

Introduction to Logarithmic Functions

  • Logarithmic functions allow for manipulation similar to algebraic functions.

  • Goal: Become proficient in using and manipulating logarithmic functions.

Properties of Logarithmic Functions

Product Property

  • Definition: If you take the log of a product, it expands to the sum of logs.

  • Formula: log_b(MN) = log_b(M) + log_b(N)

  • Example: log(4X) = log(4) + log(X)

  • Conceptual Explanation: Highlights the relationship between multiplication (M and N) and addition (logs of M and N).

Quotient Property

  • Definition: When taking the log of a quotient, it expands to the difference of logs.

  • Formula: log_b(M/N) = log_b(M) - log_b(N)

  • Example: log_b(X/10) = log_b(X) - log_b(10)

  • Contraction Method: Condensing logs while keeping the same base is essential. Example: log_b(3Y) - log_b(Z) = log_b(3Y/Z)

Power Property

  • Definition: If a log contains a number raised to a power, the exponent can be brought in front of the log as a coefficient.

  • Formula: log_b(M^p) = p * log_b(M)

  • Example: log_3(X^2) = 2 * log_3(X)

  • Use Case: This property is crucial for solving logarithmic equations and understanding their graphs.

Change of Base Formula

  • Purpose: Allows conversion of logs to a different base, which is useful for calculators that may not support all bases.

  • Formula: log_b(X) = log_k(X) / log_k(b), where k is a new base (commonly 10 or e)

  • Example: log_2(12) can be computed as log(12)/log(2) using common logs, or natural logs as ln(12)/ln(2).

  • Calculator Tip: Utilize the "log base" function if available to directly calculate logs with custom bases.

Graphical Implications of Logarithmic Manipulations

Expanding Functions

  • Example Analysis: f(x) = log_2(4X) can be expanded to log_2(4) + log_2(X).

  • Graph Interpretation: The expansion indicates a vertical shift based on the constant log value (log_2(4) = 2).

Condensing Functions

  • Example Analysis: If expanded correctly, log base information remains consistent, allowing effective graph transformations.

  • Dilation and Shifts: Understand how multiplying logs affects the vertical scaling of the graphs and their basic shifts left or right.

Practical Example: Manipulating Logs

  • Given: log base 3 of (9(2 - X))

  • Step 1: Identify common factors; here, 9 is evident as a multiplier.

  • Step 2: Apply the product property: log_3(9) + log_3(2 - X).

  • Conclusion: Calculate known log values and understand shifts in the context of the graph (e.g. vertical shifts and horizontal reflections).

Conclusion

  • Mastery of these four properties (Product, Quotient, Power, and Change of Base) is essential for handling logarithmic functions effectively.

  • Continue practicing to reinforce understanding and ability to manipulate logarithmic functions with confidence.