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 its factors. Mathematically, $\log_b(xy) = \log_b(x) + \log_b(y)$.

5 Must Know Facts For Your Next Test

1. The product rule applies to any logarithmic base, not just common logarithms (base 10).
2. It simplifies multiplication inside a logarithm into addition outside the logarithm.
3. This property is useful for solving equations and simplifying expressions involving logarithms.
4. The product rule can be used in conjunction with other logarithmic properties such as the quotient rule and power rule.
5. Understanding this rule is essential for expanding or condensing logarithmic expressions.

Review Questions

• How would you use the product rule to simplify $\log_2(8 \cdot 4)$?
• What is the simplified form of $\log_3(xy)$ using the product rule?
• Can you apply the product rule to $\ln(7 \cdot e^2)$ and simplify it?

Related terms

Quotient Rule for Logarithms: \$\$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\$\$: The logarithm of a quotient is equal to the difference of the logarithms.

Power Rule for Logarithms: \$\$\log_b(x^k) = k \cdot \log_b(x)\$\$: The logarithm of a number raised to an exponent is equal to the exponent times the logarithm of that number.

Change of Base Formula: \$\$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}\$\$: A formula used to compute a logarithm with one base using logs with another base.