Quotient rule for logarithms

Definition

The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator: $\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N$. It simplifies complex expressions involving division inside a logarithm.

5 Must Know Facts For Your Next Test

1. The quotient rule can be applied to any logarithmic base, not just base 10 or natural logs.
2. It is essential for simplifying expressions before solving equations involving logarithms.
3. The quotient rule is derived from the properties of exponents, specifically that dividing powers corresponds to subtracting their exponents.
4. This property is often used in conjunction with other logarithmic rules like the product and power rules.
5. Understanding this rule helps in converting complex fraction problems into simpler subtraction problems.

Review Questions

• How would you simplify $\log_3 \left( \frac{27}{9} \right)$ using the quotient rule?
• Explain why $\log_b M - \log_b N$ is equivalent to $\log_b \left( \frac{M}{N} \right)$
• If $\log_5 A = x$ and $\log_5 B = y$, what is $\log_5 (\frac{A}{B})$ in terms of x and y?

Related terms

Product Rule for Logarithms: $\log_b (MN) = \log_b M + \log_b N$, which states that the log of a product equals the sum of the logs.

Power Rule for Logarithms: $\log_b (M^p) = p * \log_b M$, which states that the log of a power equals the exponent times the log of the base.

Change of Base Formula: $\log_a b = \frac{\log_c b}{\log_c a}$, which allows logs to be converted between different bases.