Logarithm - Examples – How to evaluate Logarithms

• Definition: The logarithm of a positive number x to the base b is the power to which b must be raised to obtain x.
  • Written as: log_b(x) = y if and only if b^y = x.
  • Example: log₂(8) = 3 since 2³ = 8.
• Properties of Logarithms:
  1. log_b(1) = 0 for any base b.
  2. log_b(b) = 1 for any base b.
  3. log_b(b^x) = x for any base b and any positive number x.
  4. log_b(xy) = log_b(x) + log_b(y) for any base b and positive numbers x and y.
  5. log_b(x/y) = log_b(x) - log_b(y) for any base b and positive numbers x and y.
  6. log_b(xⁿ) = n * log_b(x) for any base b, positive number x, and any real number n.
• Evaluation of Logarithms: Example 1: Evaluate log₄(64)
• We need to find the power to which 4 must be raised to obtain 64.
• 4³ = 64, so log₄(64) = 3. Example 2: Evaluate log₅(1/25)
• We need to find the power to which 5 must be raised to obtain 1/25.
• 5^-2 = 1/25, so log₅(1/25) = -2. Example 3: Evaluate log₁₀(1000)
• We need to find the power to which 10 must be raised to obtain 1000.
• 10³ = 1000, so log₁₀(1000) = 3. Example 4: Evaluate log₃(1)
• We need to find the power to which 3 must be raised to obtain 1.
• 3⁰ = 1, so log₃(1) = 0. Example 5: Evaluate log₂(16)
• We need to find the power to which 2 must be raised to obtain 16.
• 2⁴ = 16, so log₂(16) = 4.

Logarithm - Examples – How to evaluate Logarithms

Slide 11:

• Example 6: Evaluate log₆(36)
  • We need to find the power to which 6 must be raised to obtain 36.
  • 6² = 36, so log₆(36) = 2.
• Example 7: Evaluate log₇(49)
  • We need to find the power to which 7 must be raised to obtain 49.
  • 7² = 49, so log₇(49) = 2.
• Example 8: Evaluate log₈(64)
  • We need to find the power to which 8 must be raised to obtain 64.
  • 8² = 64, so log₈(64) = 2.
• Example 9: Evaluate log₉(81)
  • We need to find the power to which 9 must be raised to obtain 81.
  • 9² = 81, so log₉(81) = 2.
• Example 10: Evaluate log₁₀(100000)
  • We need to find the power to which 10 must be raised to obtain 100000.
  • 10⁵ = 100000, so log₁₀(100000) = 5.

Slide 12:

• Example 11: Evaluate log₂(1/8)
  • We need to find the power to which 2 must be raised to obtain 1/8.
  • 2^-3 = 1/8, so log₂(1/8) = -3.
• Example 12: Evaluate log₃(1/27)
  • We need to find the power to which 3 must be raised to obtain 1/27.
  • 3^-3 = 1/27, so log₃(1/27) = -3.
• Example 13: Evaluate log₄(1/64)
  • We need to find the power to which 4 must be raised to obtain 1/64.
  • 4^-3 = 1/64, so log₄(1/64) = -3.
• Example 14: Evaluate log₅(1/125)
  • We need to find the power to which 5 must be raised to obtain 1/125.
  • 5^-3 = 1/125, so log₅(1/125) = -3.
• Example 15: Evaluate log₆(1/216)
  • We need to find the power to which 6 must be raised to obtain 1/216.
  • 6^-3 = 1/216, so log₆(1/216) = -3.

Slide 13:

• Example 16: Evaluate log₂(√2)
  • We need to find the power to which 2 must be raised to obtain √2.
  • 2^1/2 = √2, so log₂(√2) = 1/2.
• Example 17: Evaluate log₃(√3)
  • We need to find the power to which 3 must be raised to obtain √3.
  • 3^1/2 = √3, so log₃(√3) = 1/2.
• Example 18: Evaluate log₄(√4)
  • We need to find the power to which 4 must be raised to obtain √4.
  • 4^1/2 = √4, so log₄(√4) = 1/2.
• Example 19: Evaluate log₅(√5)
  • We need to find the power to which 5 must be raised to obtain √5.
  • 5^1/2 = √5, so log₅(√5) = 1/2.
• Example 20: Evaluate log₆(√6)
  • We need to find the power to which 6 must be raised to obtain √6.
  • 6^1/2 = √6, so log₆(√6) = 1/2.

Slide 21:

• Example 21: Evaluate log₇(√49)
  • We need to find the power to which 7 must be raised to obtain √49.
  • 7^1/2 = √49, so log₇(√49) = 1/2.
• Example 22: Evaluate log₈(√64)
  • We need to find the power to which 8 must be raised to obtain √64.
  • 8^1/2 = √64, so log₈(√64) = 1/2.
• Example 23: Evaluate log₉(√81)
  • We need to find the power to which 9 must be raised to obtain √81.
  • 9^1/2 = √81, so log₉(√81) = 1/2.
• Example 24: Evaluate log₁₀(√100)
  • We need to find the power to which 10 must be raised to obtain √100.
  • 10^1/2 = √100, so log₁₀(√100) = 1/2.
• Example 25: Evaluate log₂(2^-3)
  • We need to find the power to which 2 must be raised to obtain 2^-3.
  • 2^-3 = 1/8, so log₂(2^-3) = -3.

Slide 22:

• Example 26: Evaluate log₃(3²)
  • We need to find the power to which 3 must be raised to obtain 3².
  • 3² = 9, so log₃(3²) = 2.
• Example 27: Evaluate log₄(4^1/2)
  • We need to find the power to which 4 must be raised to obtain 4^1/2.
  • 4^1/2 = 2, so log₄(4^1/2) = 1/2.
• Example 28: Evaluate log₅(5²)
  • We need to find the power to which 5 must be raised to obtain 5².
  • 5² = 25, so log₅(5²) = 2.
• Example 29: Evaluate log₆(6²)
  • We need to find the power to which 6 must be raised to obtain 6².
  • 6² = 36, so log₆(6²) = 2.
• Example 30: Evaluate log₁₀(10^-2)
  • We need to find the power to which 10 must be raised to obtain 10^-2.
  • 10^-2 = 1/100, so log₁₀(10^-2) = -2.