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Understanding and Applying Logarithmic Concepts

Apr 26, 2024

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Lecture Notes on Logarithms

Summary

In this lecture, we explored the concept of logarithms, focusing on evaluating logarithms, using the change of base formula, expanding and condensing logarithms, solving logarithmic equations, and graphing logarithmic functions.

Evaluating Logarithms

  • Basic Concept: Logarithm asks the question: "To what power must the base be raised, to produce a given number?"
  • For log<sub>b</sub>(x):
    • Example: log<sub>2</sub>(8) = 3 because 2³ = 8.

Specific Examples:

  • log<sub>2</sub>(4) = 2
  • log<sub>2</sub>(8) = 3
  • log<sub>3</sub>(9) = 2
  • log<sub>4</sub>(16) = 2
  • log<sub>5</sub>(125) = 3
  • log<sub>6</sub>(36) = 2
  • log<sub>2</sub>(64) = 6

Change of Base Formula

  • Formula: log<sub>a</sub>(b) = (log<sub>c</sub>(b)) / (log<sub>c</sub>(a))
  • This formula allows you to change the base to any other base 'c', which can be more convenient for calculations.

Properties of Logarithms

  1. Product Rule: log<sub>b</sub>(XY) = log<sub>b</sub>(X) + log<sub>b</sub>(Y)
  2. Quotient Rule: log<sub>b</sub>(X/Y) = log<sub>b</sub>(X) - log<sub>b</sub>(Y)
  3. Power Rule: log<sub>b</sub>(X<sup>k</sup>) = k * log<sub>b</sub>(X)
  4. Expanding and Condensing Logarithms:
    • Expanding: Split the logarithmic statements using the product, quotient, or power rules.
    • Condensing: Combine logs into a single log statement using the reverse of the above rules.*

Solving Logarithmic Equations

  • Converting Logarithmic to Exponential Forms:
    • log<sub>b</sub>(X) = Y implies b^Y = X
  • Examples & Solutions:
    • Solve log<sub>2</sub>(X) = 5: 2^5 = X, hence X = 32.
    • Solve log<sub>3</sub>(2X - 3) = 2: 3² = 2X - 3, hence X = 6.

Graphing Logarithmic Functions

  • Important Point: Logarithmic functions typically include a vertical asymptote given by the set point where the log expression inside equals zero.
  • Graph Characteristics:
    • It increases or decreases, passing through key transformation points.
    • The domain typically starts from the vertical asymptote to infinity.

Quick Logarithm Review

  • Handling Different Bases:
    • log<sub>10</sub> without base mentioned means base 10.
    • Handling fractional and negative arguments within logarithmic functions.
  • Special Values:
    • log<sub>b</sub>(1) always equals 0 (because b<sup>0</sup>=1).

This comprehensively covers the introductory concepts of logarithms, further equations, properties, and their graphical representations, necessary for a foundational understanding in mathematics.

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