Properties of Logarithms: Product, Quotient, and Power Rules

Properties of Logarithms

Product, Quotient, and Power Rules of Logarithms

Logarithmic properties are essential tools for simplifying, expanding, and condensing logarithmic expressions. These properties are derived from the laws of exponents and are widely used in algebra and precalculus.

• Product Rule: The logarithm of a product is the sum of the logarithms.

• Quotient Rule: The logarithm of a quotient is the difference of the logarithms.

• Power Rule: The logarithm of a power is the exponent times the logarithm of the base.

Rule	Expression	Property	Description
Product	$\log_b(MN)$	$\log_b(M) + \log_b(N)$	Multiply terms in a log → ADD logs
Quotient	$\log_b\left(\frac{M}{N}\right)$	$\log_b(M) - \log_b(N)$	Divide terms in a log → SUBTRACT logs
Power	$\log_b(M^p)$	$p\log_b(M)$	Exponent in a log → MULTIPLY the log

Expanding and Condensing Logarithmic Expressions

Logarithmic expressions can be rewritten in expanded or condensed form using the properties above. Expanding means writing a single logarithm as a sum, difference, or multiple of logs. Condensing means combining several logarithms into a single logarithm.

Name	Expression	Property
Product Rule	$\log_b(MN)$	$\log_b(M) + \log_b(N)$
Quotient Rule	$\log_b\left(\frac{M}{N}\right)$	$\log_b(M) - \log_b(N)$
Power Rule	$\log_b(M^p)$	$p\log_b(M)$

Examples

• Expand: $\log_b(xy^2)$ $= \log_b(x) + \log_b(y^2)$ $= \log_b(x) + 2\log_b(y)$

• Condense: $2\log_b(x) + 3\log_b(y)$ $= \log_b(x^2) + \log_b(y^3)$ $= \log_b(x^2y^3)$

Additional info: When condensing logs, always apply the power rule first, then combine using the product or quotient rules as appropriate.

Evaluating Logarithms Using the Change of Base Property

Change of Base Formula

If a logarithm does not have a base that is easy to evaluate, you can use the change of base property to rewrite it in terms of common logarithms (base 10) or natural logarithms (base $e$), which are easily evaluated with a calculator.

Original	Change of Base Formula
$\log_b(N)$	$\frac{\log_a(N)}{\log_a(b)}$

Common choices for $a$ are 10 (common log) or $e$ (natural log).

Examples

• $\log_2 21 = \frac{\log_{10} 21}{\log_{10} 2} \approx 4.39$

• $\log_5 67 = \frac{\log_{10} 67}{\log_{10} 5} \approx 2.11$

• $\log_4 41 = \frac{\ln 41}{\ln 4} \approx 2.51$

• $\log_8 3709 = \frac{\ln 3709}{\ln 8} \approx 3.68$

Practice Problems

Expand or Condense Logarithmic Expressions

• Expand: $\log_b \frac{x^2 y}{z} + 2\log_b x$ $= \log_b(x^2 y) - \log_b(z) + 2\log_b(x)$ $= \log_b(x^2) + \log_b(y) - \log_b(z) + 2\log_b(x)$ $= 2\log_b(x) + \log_b(y) - \log_b(z) + 2\log_b(x)$ $= 4\log_b(x) + \log_b(y) - \log_b(z)$

• Condense: $2\log_b(x) + 3\log_b(y)$ $= \log_b(x^2) + \log_b(y^3)$ $= \log_b(x^2y^3)$

Evaluate Logarithms Using Change of Base

• $\log_3 17 = \frac{\log_{10} 17}{\log_{10} 3} \approx 2.83$

• $\log_5 67 = \frac{\log_{10} 67}{\log_{10} 5} \approx 2.11$

• $\log_4 41 = \frac{\ln 41}{\ln 4} \approx 2.51$

• $\log_8 3709 = \frac{\ln 3709}{\ln 8} \approx 3.68$