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:

1. Product Rule: log₂(x∙y) = log₂ x + log₂ y
2. Quotient Rule: log₂$x/y$ = log₂ x - log₂ y
3. 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.