Product rule for logarithms

Definition

The product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, $\log_b(xy) = \log_b(x) + \log_b(y)$.

5 Must Know Facts For Your Next Test

1. The product rule can be applied to any base of logarithm, not just base 10 or natural logarithms.
2. This property is useful in simplifying complex logarithmic expressions.
3. The rule applies only when both x and y are positive real numbers.
4. It can be derived from the properties of exponents since logarithms are essentially inverse operations of exponentiation.
5. Understanding this rule is essential for solving equations involving multiple logarithmic terms.

Review Questions

• How can you simplify $\log_2(8 \cdot 4)$ using the product rule?
• Is $\log_b(xy) = \log_b(x) + \log_b(y)$ valid for negative values of x or y? Why or why not?
• Use the product rule to simplify $\ln(3x) - \ln(3)$.

Related terms

Quotient Rule for Logarithms: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$; it expresses the logarithm of a quotient as the difference of the logarithms.

Power Rule for Logarithms: $\log_b(x^k) = k \cdot \log_b(x)$; it expresses the logarithm of an exponentiated value as a multiple of the logarithm.

Change of Base Formula: $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$; it allows conversion between different bases of logarithms.