뒤로Properties of Logarithms: Rules, Expansions, 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 for simplifying and expanding logarithmic expressions.
Definition: For positive real numbers b, M, and N with b \neq 1:
Proof: Let and . Then:
Example: Expand :
Example: Expand :
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: Expand :
Example: Expand :
Power Rule
The Power Rule states that the logarithm of a number raised to a power is equal to the product of the exponent and the logarithm of the base number.
Definition: For any real number p and positive real numbers b, M with b \neq 1:
Proof: , so
Example: Expand :
Example: Expand :
Example: Expand :
Common Mistakes with Logarithms
Students often make errors by misapplying logarithmic properties. The following are incorrect expansions:
Instead,
Correct expansion:
Expanding Logarithmic Expressions
Logarithmic expressions can be expanded using the product, quotient, and power rules. This process is useful for simplifying complex logarithms.
Example A: Expand :
Example B: Expand :
Condensing Logarithmic Expressions
Condensing involves combining multiple logarithmic terms 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 you to rewrite logarithms with any base in terms of logarithms with another base, typically base 10 (common logarithm) or base e (natural logarithm).
Definition: For positive real numbers a, b, M with a \neq 1 and b \neq 1:
Common choices: or
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 solving logarithmic equations, simplifying expressions, and graphing logarithmic functions in Precalculus.
