Logarithm Presentation Notes

Introduction to Logarithms

• Logarithm: inverse operation of exponentiation.
• Notation: log is short for logarithm.
• Example: 2³ = 8 implies log base 2 of 8 is 3.
  • Interpretation: 2 to what power equals 8? Answer is 3.

Examples and Explanations

Example 1

• Expression: log base 4 of 64 = x
• Interpretation: 4 to what power equals 64?
  • Calculation: 4³ = 64, so x = 3

Example 2

• Expression: log base 10 of 1,000,000 = ?
• Interpretation: 10 to what power equals 1,000,000?
  • Calculation: 10⁶ = 1,000,000, so x = 6

Example 3

• Expression: log base 1/2 of 1/8 = x
• Interpretation: (1/2) to what power equals 1/8?
  • Calculation: (1/2)³ = 1/8, so x = 3

Example 4

• Expression: log base x of 27 = 3
• Interpretation: x³ = 27
  • Calculation: x = 3
  • Conclusion: log base 3 of 27 = 3

Example 5 (Trick Problem)

• Expression: log base 100 of 1 = ?
• Interpretation: 100 to what power equals 1?
  • Calculation: 100⁰ = 1, so x = 0

Undefined Logarithms

Example 1

• Expression: log base 2 of 0
• Interpretation: 2 to what power equals 0?
  • Conclusion: Undefined, as no power of 2 results in 0.

Example 2

• Expression: log base 3 of -1
• Interpretation: 3 to what power equals -1?
  • Conclusion: Undefined, as no power of a positive number results in a negative number.

Additional Examples

Example 1

• Expression: log base 8 of 1/64
• Interpretation: 8 to what power equals 1/64?
  • Calculation: 8 to the (-2) power = 1/64, so x = -2
  • Note: Taking the inverse turns the answer negative.

Conclusion

• Ready for level 1 logarithm exercises.
• Future modules will cover more properties of logarithms.