Introduction to Logarithms

What is a Logarithm?

• A logarithm answers the question: How many of one number multiply together to make another number?
• Example: How many 2s multiply together to make 8?
  • 2 x 2 x 2 = 8, so the logarithm is 3.

How to Write Logarithms

• Notation:
  • log₂(8) = 3
  • The number we multiply is the "base".
  • Example Statements:
    • "The logarithm of 8 with base 2 is 3."
    • "Log base 2 of 8 is 3."
    • "The base-2 log of 8 is 3."

Components of a Logarithm

• Base: The number we are multiplying (e.g., 2 in the example above).
• Logarithm: How often the base is used in a multiplication (3 times in the example).
• Target Number: The number we want to achieve through multiplication (e.g., 8).

More Examples

• log₅(625):
  • 5 x 5 x 5 x 5 = 625, hence log₅(625) = 4.
• log₂(64):
  • 2 x 2 x 2 x 2 x 2 x 2 = 64, hence log₂(64) = 6.

Relationship with Exponents

• Exponents and Logarithms are related:
  • Exponent indicates how many times to use the base in multiplication.
  • Example: 2³ = 2 x 2 x 2 = 8.
  • Logarithms answer: "What exponent do we need?"
  • General Case: a^x = y becomes logₐ(y) = x.
• Examples:
  • log₁₀(100) = 2 (10² = 100)
  • log₃(81) = 4 (3⁴ = 81)

Common Logarithms

• Base 10 Logarithms (Common Logarithms):
  • Written as log(100) usually implies base 10.
  • Used often by engineers, usually found as "log" button on calculators.
  • Example: log(1000) = log₁₀(1000) = 3.

Natural Logarithms

• Base e Logarithms (Natural Logarithms):
  • Base e ≈ 2.71828.
  • Found as "ln" on calculators, used frequently by mathematicians.
  • Example: ln(7.389) = logₑ(7.389) ≈ 2.

Potential Confusion

• Sometimes "log" is used instead of "ln" for natural logs by mathematicians, leading to confusion.
  • Example Table:
    • log(50) can mean log₁₀(50) or logₑ(50).
    • ln(50) always means logₑ(50).

Logarithms with Decimals

• Logarithms can have decimal values (e.g., log₁₀(26) ≈ 1.41497).

Negative Logarithms

• Negative logarithms indicate division rather than multiplication.
• Examples:
  • log₈(0.125) = -1
  • log₅(0.008) = -3

Patterns in Logarithms

• Multiplying and dividing are part of a pattern with logarithms.
• Base-10 Examples Table:
  • Positive, zero, and negative logarithms follow a pattern.

Origin of "Logarithm"

• The term "Logarithm" was coined by John Napier, from Middle Latin "logarithmus" meaning "ratio-number".