Logarithm Product Rule
The logarithm product rule tells us how to handle the logarithm of a product. It states that:
\[ \log_b(xy) = \log_b(x) + \log_b(y) \]How It Works
- Multiplication Inside the Log: When you multiply two numbers \(x\) and \(y\) and then take the logarithm, the product rule lets you split that into the sum of two separate logarithms—one for \(x\) and one for \(y\).
- Derived from Exponent Rules: This rule comes from the way exponents work. Since logarithms are the inverses of exponentiation, and when you multiply numbers you add their exponents, the logarithm of a product becomes the sum of the logarithms.
Example
Suppose we have:
\[ \log_{10}(2 \times 5) \]Using the product rule:
\[ \log_{10}(2 \times 5) = \log_{10}(2) + \log_{10}(5) \]Since \(2 \times 5 = 10\) and \(\log_{10}(10) = 1\), the equation confirms:
\[ \log_{10}(2) + \log_{10}(5) = 1 \]When to Use It
- Simplifying Logarithmic Expressions: It allows you to break down a complicated logarithm into simpler parts.
- Solving Equations: When dealing with equations that involve logarithms, this property often simplifies the problem, making it easier to isolate and solve for the variable.
Important Conditions
- Positive Numbers: The rule is valid only if \(x > 0\) and \(y > 0\), because logarithms of non-positive numbers are not defined in the real number system.
- Base Restrictions: The base \(b\) must be positive and not equal to 1 (\(b > 0\) and \(b \neq 1\)).
In summary, the logarithm product rule is a powerful tool that leverages the connection between multiplication and addition to simplify and solve logarithmic expressions and equations.
Latex
\log_b(xy) = \log_b(x) + \log_b(y)