Introduction
Because logarithms are exponents, they follow exponent rules in reverse. Just as , the log of a product is the sum of the logs.
Prerequisite Connection
Recall exponent rules: multiplying bases means adding exponents, and dividing bases means subtracting exponents.
Today's Increment
We apply these rules to inputs: . This allows us to break apart complex arguments.
Why This Matters
In calculus, differentiating is a nightmare with the Chain Rule. But if we take of both sides first, it becomes a simple sum of derivatives. This technique is called Logarithmic Differentiation.
Key Concepts
The Product Rule
The log of a product is the sum of the logs.
The Quotient Rule
The log of a quotient is the difference of the logs.
Worked Examples
Example 1: Expanding Expressions (Basic)
Expand using the Product Rule.
Apply Product Rule
.
Simplify
Answer: .
Example 2: Condensing Expressions (Intermediate)
Write as a single logarithm: .
Left to Right Order
.
Combine Product
Example 3: Complex Evaluation (Advanced)
Given and , estimate .
Expand using rules
. Note that , but we don't have the Power Rule yet. We can write .
Substitute
.
Answer
Common Pitfalls
False Product Rule
. There is NO rule for the log of a sum. The logs must be summed to multiply the potential arguments.
False Quotient Rule
. The subtraction happens outside the function, or the division happens inside. Division of two separate log outputs is the Change of Base formula, not the Quotient Rule.
Real-World Application
Compression Algorithms
In information theory, calculating potential outcomes often involves multiplying huge probabilities. By taking the logarithm of these probabilities (log-likelihood), computers can transform these expensive multiplications into cheap additions, vastly speeding up data processing.
Practice Quiz
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