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How do the logarithm properties let you expand or condense a logarithmic expression?

Topic 2.12 Logarithmic Function Manipulation: rewrite logarithmic expressions using the product, quotient, power and change-of-base properties to expand or condense them.

A focused answer to AP Precalculus Topic 2.12, covering the product, quotient, power and change-of-base properties of logarithms, and how to expand a single log or condense several into one.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

The logarithm properties turn products into sums, quotients into differences, and exponents into coefficients. The product property is $\log_b(MN) = \log_b M + \log_b N$ ; the quotient property is $\log_b\frac{M}{N} = \log_b M - \log_b N$ ; the power property is $\log_b(M^k) = k\log_b M$ ; and the change-of-base property is $\log_b M = \frac{\log M}{\log b} = \frac{\ln M}{\ln b}$ . To expand, apply these left to right, breaking one logarithm into several. To condense, apply them right to left, combining several logarithms into one. These properties come directly from the exponent rules, since a logarithm is an exponent.

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What this topic is asking
The logarithm properties
Expanding a logarithm
Change of base
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What this topic is asking

The College Board (Topic 2.12) wants you to rewrite logarithmic expressions using the product, quotient, power and change-of-base properties. You must expand a single logarithm of a product, quotient or power into a sum and difference, and condense a sum and difference of logarithms back into one logarithm.

The logarithm properties

These mirror the exponent rules: multiplying numbers adds their exponents (so logs add), dividing subtracts, and raising to a power multiplies. The change-of-base rule lets you evaluate any logarithm on a calculator that only has $\log$ and $\ln$ .

Expanding a logarithm

To expand, work from the outside in: split a product into a sum, a quotient into a difference, then bring exponents down as coefficients. Roots are powers ( $\sqrt{y} = y^{1/2}$ ), so they become fractional coefficients.

Expanding and condensing

(a) Expand $\log_2\frac{8x^4}{y}$ completely. (b) Condense $3\ln a + \frac{1}{2}\ln b - \ln c$ into one logarithm.

step 1 Expand using the quotient property

$\log_2\frac{8x^4}{y} = \log_2(8x^4) - \log_2 y$ .

step 2 Split the product and bring down powers

$\log_2(8x^4) = \log_2 8 + \log_2 x^4 = 3 + 4\log_2 x$ (since $\log_2 8 = 3$ ). So the full expansion is $3 + 4\log_2 x - \log_2 y$ .

step 3 Condense the coefficients into exponents

For part (b), the power property turns coefficients into exponents: $3\ln a = \ln a^3$ and $\frac{1}{2}\ln b = \ln b^{1/2} = \ln\sqrt{b}$ .

step 4 Combine with product and quotient

$\ln a^3 + \ln\sqrt{b} - \ln c = \ln\frac{a^3\sqrt{b}}{c}$ . The sum becomes a product and the subtraction becomes a quotient.

Change of base

The change-of-base property rewrites a logarithm in any base using a base your calculator knows. To evaluate $\log_2 50$ , compute $\frac{\ln 50}{\ln 2}$ or $\frac{\log 50}{\log 2}$ . This is essential on the calculator section and for comparing logarithms of different bases.

A point worth emphasizing is that these properties only apply to a logarithm of a product, quotient or power, not to products, quotients or powers of logarithms. So $\log(MN) = \log M + \log N$ is valid, but $\log M \cdot \log N$ does not simplify, and $(\log M)^2$ is not $2\log M$ . The exam deliberately offers tempting wrong answers that misapply the properties to the outside of a logarithm, and keeping straight that the property acts on the argument, never on the logarithm as a whole, is what avoids them.

A second clarifying idea is that expanding and condensing are exact inverses, so you can always check your work by reversing it. After condensing into $\log\frac{5x^2}{y}$ , expanding it should return $2\log x - \log y + \log 5$ . Using the reverse direction as a self-check catches sign errors and misplaced coefficients, which are the most common slips when several properties are chained together.

Try this

Q1. Expand $\log(5x)$ . [1 point]

Cue. Product property: $\log(5x) = \log 5 + \log x$ .

Q2. Condense $\log x - 3\log y$ into one logarithm. [1 point]

Cue. $3\log y = \log y^3$ , then quotient: $\log\frac{x}{y^3}$ .

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2023 (style)1 marksSection I, Part A (multiple choice, no calculator). Which expression equals

\log(xy^2)

? (A)

\log x + 2\log y

(B)

\log x \cdot 2\log y

(C)

2\log(xy)

(D)

\log x + \log y^{2} - \log 1

Show worked answer →

The correct answer is (A), $\log x + 2\log y$ .

The product property splits $\log(xy^2) = \log x + \log y^2$ , and the power property brings the exponent down: $\log y^2 = 2\log y$ . So $\log(xy^2) = \log x + 2\log y$ . Choice (B) wrongly multiplies the logs, which is not a property.

AP 2024 (style)3 marksSection II (free response, no calculator). (a) Expand

\ln\frac{x^3}{\sqrt{y}}

completely. (b) Condense

2\log x - \log y + \log 5

into a single logarithm.

Show worked answer →

A 3-point question on expanding and condensing logs.

(a) Expand (2 points): $\ln\frac{x^3}{\sqrt{y}} = \ln x^3 - \ln y^{1/2} = 3\ln x - \frac{1}{2}\ln y$ , using the quotient property then the power property.
(b) Condense (1 point): $2\log x = \log x^2$ and $\log 5$ adds (product), $-\log y$ subtracts (quotient), giving $\log\frac{5x^2}{y}$ .

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Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)