Understanding Logarithms and Their Properties
This page provides a comprehensive overview of logarytmy (logarithms) and their fundamental properties. It begins by introducing the concept of logarithms and their relationship to exponents.
Definition: A logarithm is the power to which a base must be raised to produce a given number. For example, log₁₂ c = 6 means 12⁶ = c.
The page then focuses on the logarytm dziesiętny (decimal logarithm), which is a logarithm with base 10. This is expressed as:
Example: log₁₀ x = c is equivalent to 10ᶜ = x
Several key properties and operations of logarithms are presented:
- Product Rule: log₂(x∙y) = log₂ x + log₂ y
- Quotient Rule: log₂(x/y) = log₂ x - log₂ y
- Power Rule: log₂(xᵐ) = m ∙ log₂ x
Highlight: These rules are fundamental for simplifying and solving logarithmic expressions.
The document also introduces special cases of logarithms:
Vocabulary:
- log₂ 1 = 0
- log₂ a = 1 (where a is the base)
Finally, the page presents the change of base formula, which is crucial for converting between logarithms with different bases:
Formula: log₂ c = (log₁₀ c) / (log₁₀ 2)
This formula allows for the calculation of any logarithm using the common (base 10) logarithm, which is particularly useful when using calculators or tables.
Example: To calculate log₂ 8, you can use (log₁₀ 8) / (log₁₀ 2) ≈ 3
The page concludes by emphasizing that these properties and formulas apply to all positive real numbers and bases greater than 0 and not equal to 1.
