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Understanding Logarithmic Properties

May 8, 2025

Properties of Logarithms

Overview

  • Power Rule: For log(a^n), you can move the exponent n in front, resulting in n * log(a).
  • Product Rule: log(a * b) is equal to log(a) + log(b).
  • Quotient Rule: log(a / b) is equal to log(a) - log(b).

Examples

Simplifying Logs

  1. Example 1:

    • Expression: log base 5 of 5^7
    • Simplification: Move 7 to the front: 7 * log base 5 of 5
    • Result: Since log base 5 of 5 is 1, the result is 7 * 1 = 7.

  2. Example 2:

    • Expression: log base 2 of 8^5
    • Simplification: Move 5 to the front: 5 * log base 2 of 8
    • Calculation: log base 2 of 8 is 3 (since 2^3 = 8)
    • Result: 5 * 3 = 15

Using the Product Rule

  1. Example 3:

    • Expression: log base 2 of (16 * 8)
    • Simplification: log base 2 of 16 + log base 2 of 8
    • Calculation:
      • log base 2 of 16 is 4 (since 2^4 = 16)
      • log base 2 of 8 is 3
    • Result: 4 + 3 = 7

  2. Example 4:

    • Expression: log base 3 of (27 * 81)
    • Simplification: log base 3 of 27 + log base 3 of 81
    • Calculation:
      • log base 3 of 27 is 3 (since 3^3 = 27)
      • log base 3 of 81 is 4 (since 3^4 = 81)
    • Result: 3 + 4 = 7

Using the Quotient Rule

  1. Example 5:

    • Expression: log base 4 of (256 / 64)
    • Simplification: log base 4 of 256 - log base 4 of 64
    • Calculation:
      • log base 4 of 256 is 4 (since 4^4 = 256)
      • log base 4 of 64 is 3 (since 4^3 = 64)
    • Result: 4 - 3 = 1

  2. Example 6:

    • Expression: log base 2 of (128 / 8)
    • Simplification: log base 2 of 128 - log base 2 of 8
    • Calculation:
      • log base 2 of 128 is 7 (since 2^7 = 128)
      • log base 2 of 8 is 3
    • Result: 7 - 3 = 4

Complex Example

  1. Example 7:
    • Expression: log base 2 of (128 * 64 / 8 * 16)^5
    • Process:
      • Move 5 to the front and distribute: 5 * (log base 2 of 128 + log base 2 of 64 - log base 2 of 8 - log base 2 of 16)
    • Calculation:
      • log base 2 of 128 is 7
      • log base 2 of 64 is 6
      • log base 2 of 8 is 3
      • log base 2 of 16 is 4
      • Combine: 7 + 6 - 3 - 4 = 6
    • Result: 5 * 6 = 30

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