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: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$, where $b$ is the base.

5 Must Know Facts For Your Next Test

1. It simplifies complex expressions involving division inside a logarithm.
2. Works for any positive real numbers M and N, and any positive base b ≠ 1.
3. Used in conjunction with other logarithmic properties like the product and power rules.
4. Helps in solving exponential and logarithmic equations by breaking them down.
5. Essential for understanding how to manipulate expressions in algebraic problems.

Review Questions

• How do you express $\log_2\left(\frac{8}{4}\right)$ using the quotient rule?
• Explain why $\log_3\left(\frac{27}{9}\right) = \log_3(27) - \log_3(9)$ is valid.
• What are the steps to apply the quotient rule for simplifying $\log_b\left(\frac{x^2+1}{x-1}\right)$?

Related terms

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

Power Rule for Logarithms: $\log_b(M^p) = p \cdot \log_b(M)$, which expresses the log of a power as a multiple of the exponent.

Change of Base Formula: $\log_b(M) = \frac{\log_k(M)}{\log_k(b)}$, allowing conversion between different log bases.