I. Definition and Examples of Logarithms
Logarithms are mathematical operations that determine the exponent to which a base must be raised to produce a given number. This concept is fundamental in various mathematical and real-world applications.
Definition: The logarithm of a number P to the base a is the exponent x to which the base a must be raised to obtain P. It is written as log₂ P = x, which means a^x = P.
The base of a logarithm must be positive and not equal to 1 (a > 0, a ≠ 1). This restriction ensures that logarithms are well-defined and have unique solutions.
Example: log₂ 8 = 3 because 2³ = 8. This means that 3 is the exponent to which 2 must be raised to obtain 8.
Additional examples illustrate the concept of logarithms with different bases and values:
- log₂ (1/8) = -3, as 2^(-3) = 1/8
- log₁₀ 10000 = 4, since 10⁴ = 10000
- log₁₀ 0.0001 = -4, because 10^(-4) = 1/10000 = 0.0001
Highlight: Understanding these examples is crucial for mastering the concept of logarithms and applying them to solve complex problems.
II. Properties of Logarithms
Logarithms have several important properties that simplify calculations and problem-solving:
- Uniqueness: Different numbers have different logarithms (if P ≠ Q, then log₂ P ≠ log₂ Q)
- Logarithm of the base: The logarithm of the base itself is always 1 (log₂ a = 1)
- Logarithm of 1: The logarithm of 1 is always 0, regardless of the base (log₂ 1 = 0)
- Product rule: The logarithm of a product is the sum of the logarithms of the factors (log₂ (PQ) = log₂ P + log₂ Q)
Vocabulary: The product rule is a fundamental property that allows us to simplify complex logarithmic expressions involving multiplication.
