IndietroProperties of Logarithms: Product, Quotient, Power, Expansion, Condensing, and Change-of-Base
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Properties of Logarithms
Product Rule
The Product Rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. This property is fundamental in simplifying and expanding logarithmic expressions.
Definition: For positive real numbers b, M, and N with b \neq 1:
Proof: Let and . Then:
Example:
Example:
Quotient Rule
The Quotient Rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.
Definition: For positive real numbers b, M, and N with b \neq 1:
Proof: Let and . Then:
Example:
Example:
Power Rule
The Power Rule allows us to bring exponents in the argument of a logarithm out as a coefficient.
Definition: For any real number p and positive real numbers b, M with b \neq 1:
Proof: , so
Example:
Example:
Example:
Common Mistakes
Students often make errors by misapplying logarithm properties. The following are incorrect expansions:
Correct:
Correct:
Expanding Logarithmic Expressions
Logarithmic expressions can be expanded using the product, quotient, and power rules.
Example A:
Example B:
Condensing Logarithmic Expressions
Multiple logarithmic terms can be combined into a single logarithm using the properties above.
Example a:
Example b:
Example c:
Example d:
Example e:
Change-of-Base Property
The Change-of-Base Property allows us to rewrite logarithms with any base in terms of logarithms with another base, typically base 10 or base e. This is especially useful for evaluating logarithms on calculators.
Definition: For positive real numbers a, b, M with a \neq 1, b \neq 1:
Common choices: (common logarithm), (natural logarithm)
or
Example:
To graph , use or
Summary Table: Logarithm Properties
Property | Formula | Example |
|---|---|---|
Product Rule | ||
Quotient Rule | ||
Power Rule | ||
Change-of-Base |
Additional info: These properties are essential for simplifying, expanding, and evaluating logarithmic expressions in precalculus and calculus. Mastery of these rules is foundational for solving logarithmic equations and understanding exponential and logarithmic functions.
