Algebra 2 Quiz: Using Logarithm Properties

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Using Logarithm PropertiesQuestion 1 of 20

Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to evaluate without a calculator: $\log_3\left(\frac{81}{9}\right).$

\tfrac{1}{2}

0

2

4

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Algebra 2 Quiz

Algebra 2 Quiz: Using Logarithm Properties

Practice Using Logarithm Properties in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Using Logarithm Properties, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$\tfrac{1}{2}$
$0$
$2$ (correct answer)
$4$

Explanation: This question tests your understanding of the quotient property of logarithms and your ability to evaluate logs by recognizing powers of the base. The quotient property states that log_b(x/y) = log_b(x) - log_b(y), which means you can split a fraction inside a log into a difference, or evaluate it directly if you recognize the result as a power of the base. To evaluate log_3(81/9): First, calculate the fraction: 81/9 = 9. Then evaluate log_3(9): Since 9 = 3², we have log_3(9) = log_3(3²) = 2. Alternatively, using the quotient property: log_3(81/9) = log_3(81) - log_3(9) = log_3(3⁴) - log_3(3²) = 4 - 2 = 2. Choice C correctly gives 2 as the answer, whether you simplify first or use the quotient property. Choice A (1/2) would mean 3^(1/2) = 81/9, but 3^(1/2) = √3 ≈ 1.73, not 9. The key insight is recognizing powers of 3: 3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81. Evaluation strategy for quotients: (1) You can simplify the fraction first if possible (81/9 = 9), then evaluate the simpler log, (2) Or apply quotient property and evaluate each log separately, (3) Both methods give the same answer—choose based on which recognition is easier. Mental math tip: memorize small powers of common bases (2, 3, 5, 10) to quickly evaluate many logarithms without a calculator!

Question 2

Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to expand the expression as a sum and/or difference of logarithms: $\log_3\left(\frac{x^2y}{z^3}\right).$

$\log_3(x^2)+\log_3(y)+\log_3(z^3)$
$2\log_3(x)+\log_3(y)-3\log_3(z)$ (correct answer)
$\log_3(x)+\log_3(y)-\log_3(z)$
$\log_3(x^2+y)-\log_3(z^3)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you expand complex logarithmic expressions into sums and differences. The three logarithm properties come from exponent rules and work because logarithms are exponents: (1) Product property: log_b(xy) = log_b(x) + log_b(y) because multiplying numbers means adding their exponents, (2) Quotient property: log_b(x/y) = log_b(x) - log_b(y) because dividing means subtracting exponents, (3) Power property: log_b(

x^p

) = p log_b(x) because raising to power means multiplying exponents. To expand log_3((

x^2

y)/z^3), first apply the quotient property: log_3(

x^2

y) - log_3(

z^3

), then apply the product property to the numerator: log_3(

x^2

) + log_3(y) - log_3(

z^3

), and finally the power property: 2 log_3(x) + log_3(y) - 3 log_3(z). Choice B correctly applies these properties in sequence to fully expand the expression into a sum and difference of simpler logs with coefficients. Choice D incorrectly combines x^2 + y inside a log instead of using the product property for x^2 y, remember there's no property for log of a sum—it's a common mix-up with the product rule! When expanding logarithms, start with the outermost operation like quotients or overall powers, then break down products and individual powers step by step. This systematic approach ensures you don't miss any terms—keep practicing, you've got this!

Question 3

Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y)$ , $\log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y)$ , and $\log_b(x^p)=p\log_b(x)$ to expand the expression: $\log_5\left(\frac{x^3\sqrt{y}}{z^2}\right).$ Write your answer as a sum/difference of logarithms.

$3\log_5(x)+\frac{1}{2}\log_5(y)-2\log_5(z)$ (correct answer)
$\log_5(x^3)+\log_5(\sqrt{y})+\log_5(z^2)$
$\log_5(x^3)+\log_5(\sqrt{y})-\log_5(z)$
$\log_5\left(\frac{x+y}{z}\right)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you expand complex logarithmic expressions into sums and differences. The three logarithm properties come from exponent rules and work because logarithms ARE exponents: (1) Product property:

\log_b(xy) = \log_b(x) + \log_b(y)

because multiplying numbers means adding their exponents, (2) Quotient property:

\log_b(x/y) = \log_b(x) - \log_b(y)

because dividing means subtracting exponents, (3) Power property:

\log_b(x^p) = p \log_b(x)

because raising to power means multiplying exponents. These properties transform multiplication into addition, division into subtraction, and exponentiation into multiplication—powerful simplification tools! To expand

\log_5((x^3 \sqrt{y})/z^2)

using properties: (1) Apply quotient property first:

\log_5(x^3 \sqrt{y}) - \log_5(z^2)

because the overall structure is a fraction. (2) Apply product property to first log:

\log_5(x^3) + \log_5(\sqrt{y})

because

x^3 \sqrt{y}

is a product. (3) Apply power property to each:

3\log_5(x) + \frac{1}{2}\log_5(y) - 2\log_5(z)

because

x^3

has exponent 3,

\sqrt{y}

y^{1/2}

, and

z^2

has exponent 2. Final expanded form:

3\log_5(x) + \frac{1}{2}\log_5(y) - 2\log_5(z)

. Choice A correctly applies the logarithm properties in the right order to expand into sum and difference of simpler logs. Choice B incorrectly keeps the logs condensed without fully expanding using the power property, leaving exponents inside instead of bringing them out as coefficients—remember to apply power rule to complete the expansion! Expanding logarithm strategy: (1) Identify overall structure: is it product, quotient, or power? Start with outermost operation. (2) Apply matching property: multiplication becomes +, division becomes -, exponent becomes coefficient. (3) Work from outside in: handle fraction first (quotient property), then products within (product property), then exponents (power property). (4) Continue until each log has single variable with coefficient. You're doing great—keep practicing this step-by-step approach to master log expansions!

Question 4

Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to expand the expression $\log\left(\frac{(mn)^4}{p}\right)$ (common log base 10) into a sum/difference of logs with no exponents inside the logs.

$4\log(mn)-\log(p)$
$4\log(m)+4\log(n)-\log(p)$ (correct answer)
$\log(m)+\log(n)-\log(p^4)$
$\log(m+n)^4-\log(p)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you expand complex logarithmic expressions into sums and differences with no exponents inside the logs. The three properties are: (1) Product property: log(xy) = log(x) + log(y), (2) Quotient property: log(x/y) = log(x) - log(y), (3) Power property: log(

x^p

) = p·log(x)—this one is crucial for removing exponents from inside logs. To expand log((mn)⁴/p): First apply the quotient property to the overall fraction: log((mn)⁴) - log(p). Next apply the power property to the first term: 4log(mn) - log(p), bringing the exponent 4 down as a coefficient. Then apply the product property inside: 4[log(m) + log(n)] - log(p). Finally distribute the 4: 4log(m) + 4log(n) - log(p). Choice B correctly shows the fully expanded form with no exponents inside any logarithm and the coefficient 4 distributed to both m and n. Choice A incorrectly leaves mn together inside the log instead of separating them—when you have 4log(mn), you must apply the product property to get 4log(m) + 4log(n), not just 4log(mn). The key insight is that (mn)⁴ = m⁴n⁴, so log((mn)⁴) = log(m⁴n⁴) = log(m⁴) + log(n⁴) = 4log(m) + 4log(n). Expansion strategy for removing all exponents: (1) Handle the overall structure first (quotient here), (2) Bring down any exponents as coefficients using power property, (3) Expand any products or quotients inside, (4) Distribute coefficients if needed. The goal is to have each log contain only a single variable with no exponent, with any powers appearing as coefficients in front!

Question 5

Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to write as a single logarithm: $\log_2(5)+\log_2(3)-\log_2(4).$

$\log_2\left(\dfrac{15}{4}\right)$ (correct answer)
$\log_2\left(\dfrac{5}{12}\right)$
$\log_2(5+3-4)$
$\log_2\left(\dfrac{15}{8}\right)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you condense sums and differences of logarithms into a single logarithm. The properties for condensing are: (1) Product property backwards: log_b(x) + log_b(y) = log_b(xy), so addition becomes multiplication, (2) Quotient property backwards: log_b(x) - log_b(y) = log_b(x/y), so subtraction becomes division. To condense log_2(5) + log_2(3) - log_2(4): First combine the addition: log_2(5) + log_2(3) = log_2(5·3) = log_2(15). Then handle the subtraction: log_2(15) - log_2(4) = log_2(15/4). Choice A correctly shows the condensed form as log_2(15/4), a single logarithm of the fraction 15/4. Choice C incorrectly tries to combine using 5 + 3 - 4 inside the log, but there is NO logarithm property that allows log(a) + log(b) - log(c) = log(a + b - c). You must use multiplication for addition of logs and division for subtraction of logs! Condensing strategy: (1) Work left to right combining two logs at a time, (2) Addition of logs → multiply the arguments, (3) Subtraction of logs → divide the arguments, (4) Never add or subtract the arguments directly. The pattern to remember: plus outside means times inside, minus outside means divide inside!

Question 6

Using the properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x),$ write the expression as a single logarithm (condense): $2\log_5(a)-\tfrac{1}{2}\log_5(b)+\log_5(c).$

$\log_5\left(\frac{a^2c}{b^2}\right)$
$\log_5(a^2)+\log_5(c)-\log_5(\sqrt{b})$
$\log_5\left(\frac{a^2c}{\sqrt{b}}\right)$ (correct answer)
$\log_5\left(\frac{a^2\sqrt{b}}{c}\right)$

log_b(xy) = log_b(x) + log_b(y)

because multiplying numbers means adding their exponents, (2) Quotient property:

log_b(x/y) = log_b(x) - log_b(y)

because dividing means subtracting exponents, (3) Power property:

log_b(x^p) = p\cdot log_b(x)

because raising to power means multiplying exponents. To condense

2log_5(a) - (1/2)log_5(b) + log_5(c)

, we work backwards: (1) Apply power property backwards:

2log_5(a)

becomes

log_5(a^2)

and

(1/2)log_5(b)

becomes

log_5(b^(1/2)) = log_5(\sqrt{b})

, (2) Rewrite:

log_5(a^2) - log_5(\sqrt{b}) + log_5(c)

, (3) Apply quotient property backwards to first two:

log_5(a^2/\sqrt{b}) + log_5(c)

, (4) Apply product property backwards:

log_5((a^2/\sqrt{b})\cdot c) = log_5(a^2c/\sqrt{b})

. Choice A correctly condenses the expression into a single logarithm by applying properties in reverse order. Choice B incorrectly has

\sqrt{b}

in the numerator instead of denominator—remember that subtraction of logs means division, so

-log_5(\sqrt{b})

puts

\sqrt{b}

in the denominator! Condensing strategy (reverse process): (1) Use power property backwards: coefficient in front becomes exponent (

2log(x)

becomes

log(x^2)

), (2) Identify subtractions:

log(a) - log(b)

condenses to

log(a/b)

using quotient property backwards, (3) Identify additions:

log(a) + log(b)

condenses to

log(ab)

using product property backwards, (4) Combine all: the result should be single log of one big expression. Work step-by-step combining two logs at a time, and be careful with signs—minus means divide!

Question 7

Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to simplify the expression: $\frac{\log_5(125x)}{\log_5(5)}.$

$\log_5(125x)$
$3+\log_5(x)$ (correct answer)
$\log_5(3x)$
$\dfrac{3}{\log_5(x)}$

Explanation: This question tests your understanding of logarithm properties and the special case when the denominator is log_b(b) = 1, which allows for elegant simplification. The key insight is that log_5(5) = 1 because 5¹ = 5, so the expression becomes log_5(125x)/1 = log_5(125x). To simplify log_5(125x)/log_5(5): First recognize that log_5(5) = 1, so we have log_5(125x)/1 = log_5(125x). Next, apply the product property: log_5(125x) = log_5(125) + log_5(x). Then evaluate log_5(125): Since 125 = 5³, we have log_5(125) = log_5(5³) = 3. Therefore: log_5(125x) = 3 + log_5(x). Choice B correctly shows the simplified form as 3 + log_5(x), recognizing that 125 = 5³ contributes a value of 3. Choice A just writes log_5(125x) without simplifying—while not wrong, it misses the opportunity to evaluate log_5(125) = 3. Choice C incorrectly suggests log_5(125) = log_5(3), but 125 = 5³, not 3·5. Simplification strategy when log_b(b) appears: (1) Always remember log_b(b) = 1 for any base b, (2) This often creates opportunities to simplify further, (3) Look for perfect powers of the base to evaluate (like 125 = 5³ here), (4) Express the final answer in simplest form by evaluating any logs of perfect powers. This type of problem rewards recognizing special values and powers of the base!

Question 8

If $\log_5(a) + 3\log_5(b) - \log_5(c) = \log_5(20)$ , which equation is equivalent using logarithm properties?

$ab^3 - c = 20$
$a + 3b - c = 20$
$\frac{ab^3}{c} = 5^{20}$
$\frac{ab^3}{c} = 20$ (correct answer)

Explanation: When you encounter logarithmic equations with multiple terms, your goal is to use logarithm properties to simplify the left side and then apply the definition of logarithms to eliminate the log notation entirely. Start by applying the key logarithm properties to the left side of

\log_5(a) + 3\log_5(b) - \log_5(c) = \log_5(20)

. First, use the power property:

3\log_5(b) = \log_5(b^3)

. Next, apply the addition property (logs of products) and subtraction property (logs of quotients):

\log_5(a) + \log_5(b^3) - \log_5(c) = \log_5\left(\frac{a \cdot b^3}{c}\right)

. Now your equation becomes

\log_5\left(\frac{ab^3}{c}\right) = \log_5(20)

. Since both sides have the same base logarithm, the arguments must be equal:

\frac{ab^3}{c} = 20

. Looking at the wrong answers: Choice A (

ab^3 - c = 20

) incorrectly treats subtraction of logs as regular subtraction instead of division. Choice B (

a + 3b - c = 20

) makes the fundamental error of treating logarithms as if they follow normal arithmetic rules, ignoring all logarithm properties. Choice C (

\frac{ab^3}{c} = 5^{20}

) correctly applies logarithm properties but fails to recognize that when

\log_5(x) = \log_5(20)

, then

x = 20

, not

x = 5^{20}

. Remember: when working with logarithmic equations, always consolidate terms using properties first, then use the fact that if

\log_b(x) = \log_b(y)

, then

x = y

Question 9

If $\log_3(2x + 1) + \log_3(x - 2) = \log_3(13)$ , which expression is equivalent to the left side of the equation after applying logarithm properties?

$\log_3((2x + 1)(x - 2))$ (correct answer)
$\log_3(3x - 1)$
$\log_3(2x^2 - 3x - 2)$
$\log_3(x^2 - 4)$

Explanation: Using the product property of logarithms,

\log_b(m) + \log_b(n) = \log_b(mn)

, we get

\log_3(2x + 1) + \log_3(x - 2) = \log_3((2x + 1)(x - 2))

. Choice B incorrectly adds the arguments. Choice C expands the product but doesn't maintain the logarithm form. Choice D incorrectly assumes

(2x + 1) = (x + 2)

Question 10

If $\log_6(m) = 2$ and $\log_6(n) = -1$ , what is $\log_6\left(\frac{m^2\sqrt{n}}{36}\right)$ ?

$2.5$
$3$
$1.5$ (correct answer)
$2$

Explanation: Using logarithm properties:

\log_6\left(\frac{m^2\sqrt{n}}{36}\right) = \log_6(m^2) + \log_6(\sqrt{n}) - \log_6(36) = 2\log_6(m) + \frac{1}{2}\log_6(n) - \log_6(6^2) = 2(2) + \frac{1}{2}(-1) - 2 = 4 - 0.5 - 2 = 1.5

. Choice A results from incorrectly calculating

4 - 0.5 - 1 = 2.5

(using

\log_6(36) = 1

). Choice B results from omitting the

\log_6(36)

term entirely. Choice D results from incorrectly calculating

\log_6(36) = 1

instead of 2.

Question 11

Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ and the change-of-base formula $\log_b(x)=\dfrac{\ln(x)}{\ln(b)}$ to rewrite $\log_4(9)$ in terms of natural logs.

$\dfrac{\ln(4)}{\ln(9)}$
$\dfrac{\ln(9)}{\ln(4)}$ (correct answer)
$\ln(36)$
$\dfrac{\ln(9)}{4}$

Explanation: This question tests your understanding of the change-of-base formula, which connects logarithms of different bases and is essential for calculator evaluation since most calculators only have ln and log₁₀ buttons. The change-of-base formula states that log_b(x) = ln(x)/ln(b) = log(x)/log(b), allowing you to express any logarithm in terms of natural logs (ln) or common logs (log₁₀). To rewrite log_4(9) using natural logs: Apply the change-of-base formula directly with b = 4 and x = 9: log_4(9) = ln(9)/ln(4). This fraction form asks "what power of 4 gives 9?" which equals "ln(9) divided by ln(4)." Choice B correctly shows ln(9)/ln(4), which is the direct application of the change-of-base formula. Choice A incorrectly flips the fraction to ln(4)/ln(9), but this would represent log_9(4), not log_4(9)—the base goes in the denominator! The formula makes intuitive sense: to find what power of 4 gives 9, you divide 9's natural log by 4's natural log. Change-of-base strategy and verification: (1) The argument (9) goes in the numerator, the base (4) goes in the denominator, (2) You can verify: if log_4(9) = k, then 4^k = 9, so k·ln(4) = ln(9), thus k = ln(9)/ln(4), (3) This formula is invaluable for calculator work and for proving properties across different bases. Remember the pattern: log_base(argument) = ln(argument)/ln(base)—argument up top, base below!

Question 12

Expand the expression using the product, quotient, and power properties: $\ln\left(\frac{a^3\sqrt{b}}{c}\right).$

$3\ln(a)+\tfrac{1}{2}\ln(b)-\ln(c)$ (correct answer)
$\ln(a)+\ln(b)-\ln(c)$
$\ln(a^3)+\ln(\sqrt{b})+\ln(c)$
$\ln(a^3+b^{1/2})-\ln(c)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you expand complex logarithmic expressions into sums and differences. The three logarithm properties come from exponent rules and work because logarithms are exponents: (1) Product property: ln(xy) = ln(x) + ln(y), (2) Quotient property: ln(x/y) = ln(x) - ln(y), (3) Power property: ln(

x^p

) = p ln(x). To expand ln( (a^3 sqrt(b)) / c ), first quotient: ln(a^3 sqrt(b)) - ln(c), then product: ln(

a^3

) + ln(sqrt(b)) - ln(c), and power: 3 ln(a) + (1/2) ln(b) - ln(c). Choice B correctly applies the properties to fully expand with coefficients. Choice D incorrectly keeps a^3 + b^{1/2} inside a log—there's no sum property, it's for products! Expand by addressing quotients first, then products, then powers, working outside in. You're doing wonderfully—keep applying this step-by-step!

Question 13

Use the properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to expand $\log\left(\frac{(mn^3)^2}{p}\right)$ , where $\log$ is base 10. Write your answer as a sum/difference of logs with coefficients.

$2\log(m)+6\log(n)-\log(p)$ (correct answer)
$\log(m)+3\log(n)-\log(p)$
$2\log(m)+3\log(n)-2\log(p)$
$\log(m+ n^3)-\log(p)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you expand complex logarithmic expressions into sums and differences or condense multiple logs into a single logarithm. The three logarithm properties come from exponent rules: (1) Product property: log_b(xy) = log_b(x) + log_b(y), (2) Quotient property: log_b(x/y) = log_b(x) - log_b(y), (3) Power property: log_b(

x^p

) = p·log_b(x). To expand log((mn^3)^2/p): (1) Apply quotient property first: log((mn^3)^2) - log(p), (2) Apply power property to (mn^3)^2: log((mn^3)^2) = 2log(

mn^3

), so we have 2log(

mn^3

) - log(p), (3) Apply product property inside: 2[log(m) + log(

n^3

)] - log(p) = 2log(m) + 2log(

n^3

) - log(p), (4) Apply power property to n^3: 2log(m) + 2·3log(n) - log(p) = 2log(m) + 6log(n) - log(p). Choice A correctly applies all properties in the right order, carefully handling the outer exponent 2 that applies to the entire product mn^3. Choice D incorrectly suggests log(m +

n^3

), but there is NO property for log(x + y)—the product property applies to multiplication log(xy), not addition! Expanding with nested exponents: (1) Handle outer operations first (quotient in this case), (2) When you see (expression)^p, use power property to get p·log(expression), (3) Then expand what's inside the log using product property, (4) Distribute any coefficients from outer power property to all terms inside. The key insight: (mn^3)^2 = m^2n^6, so the final coefficients are 2 for m and 6 for n!

Question 14

Expand using the properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x):$ $\log\left(\frac{(ab)^3}{c^2d}\right).$ Write the result as a sum/difference of common logs with coefficients.

$3\log(a)+3\log(b)-2\log(c)-\log(d)$ (correct answer)
$\log(a)+\log(b)-\log(c)-\log(d)$
$3\log(ab)-\log(c^2d)$
$\log(a^3+b^3)-\log(c^2d)$

x^p

) = p·log_b(x). To expand log((ab)^3/(

c^2d

)): (1) Apply quotient property first: log((ab)^3) - log(

c^2d

), (2) Apply power property to (ab)^3: 3log(ab) - log(

c^2d

), (3) Apply product property to both parts: 3[log(a) + log(b)] - [log(

c^2

) + log(d)] = 3log(a) + 3log(b) - log(

c^2

) - log(d), (4) Apply power property to c^2: 3log(a) + 3log(b) - 2log(c) - log(d). Choice A correctly expands with coefficient 3 for both a and b (from the cubed term) and coefficient 2 for c. Choice D incorrectly writes log(

a^3

b^3

), but (ab)^3 ≠ a^3 + b^3—the cube applies to the product ab, giving a^3b^3, not their sum! Expanding with grouped bases: When you see (ab)^p, the exponent p applies to the entire product: (ab)^p = a^p·b^p, so log((ab)^p) = p·log(ab) = p[log(a) + log(b)] = p·log(a) + p·log(b). The coefficient p distributes to each term inside!

Question 15

$\tfrac12\log_7(x)+3\log_7(y)-2\log_7(w)$ (correct answer)
$\log_7(x)+\log_7(y^3)-\log_7(w^2)$
$\tfrac12\log_7(x)+3\log_7(y)+2\log_7(w)$
$\log_7(\sqrt{x}+y^3)-\log_7(w^2)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you expand complex logarithmic expressions containing roots and powers into sums and differences of simpler logs. The three properties are: (1) Product property: log_b(xy) = log_b(x) + log_b(y), (2) Quotient property: log_b(x/y) = log_b(x) - log_b(y), (3) Power property: log_b(

x^p

) = p·log_b(x), and remember that √x = x^(1/2). To expand log_7(√x·y³/w²): First apply quotient property to the fraction: log_7(√x·y³) - log_7(w²). Next apply product property to the numerator: log_7(√x) + log_7(y³) - log_7(w²). Then apply power property to each term, remembering √x = x^(1/2): (1/2)log_7(x) + 3log_7(y) - 2log_7(w). Choice A correctly shows the fully expanded form with the square root converted to the coefficient 1/2, the cube to coefficient 3, and the square to coefficient 2. Choice D incorrectly tries to write √x + y³ inside a single log, but this violates the product property—log(√x·y³) expands to log(√x) + log(y³), not log(√x + y³). There's no logarithm property for addition inside! The key insight for roots: √x = x^(1/2), so log(√x) = log(x^(1/2)) = (1/2)log(x). Expansion strategy with roots and powers: (1) Recognize roots as fractional exponents (√x = x^(1/2), ∛x = x^(1/3)), (2) Apply properties systematically from outside in, (3) All exponents become coefficients, including fractions from roots. This creates a clean expanded form where each log contains a single variable and all powers appear as coefficients out front!

Question 16

Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to expand the expression $\log_3\left(\frac{x^2y}{z^3}\right)$ into a sum/difference of logs.

$\log_3(x^2)+\log_3(y)-\log_3(z^3)$
$2\log_3(x)+\log_3(y)-3\log_3(z)$ (correct answer)
$\log_3(x)+\log_3(y)-\log_3(z)+5$
$\log_3(x+y)-\log_3(z^3)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you expand complex logarithmic expressions into sums and differences of simpler logs. The three logarithm properties come from exponent rules and work because logarithms ARE exponents: (1) Product property: log_b(xy) = log_b(x) + log_b(y) because multiplying numbers means adding their exponents (b^m × b^n = b^(m+n)), (2) Quotient property: log_b(x/y) = log_b(x) - log_b(y) because dividing means subtracting exponents (b^m ÷ b^n = b^(m-n)), (3) Power property: log_b(

x^p

) = p·log_b(x) because raising to power means multiplying exponents ((b^m)^p = b^(mp)). To expand log_3(x²y/z³): First apply quotient property to the overall fraction structure: log_3(x²y) - log_3(z³). Then apply product property to the first term: log_3(x²) + log_3(y) - log_3(z³). Finally apply power property to bring exponents down: 2log_3(x) + log_3(y) - 3log_3(z). Choice B correctly applies all three properties in the right order, expanding the complex log into a sum and difference of simple logs with coefficients. Choice A incorrectly leaves the exponents inside the logs instead of bringing them down as coefficients—the power property says log_3(x²) = 2log_3(x), not just log_3(x²). Expanding strategy: (1) Identify overall structure (quotient here), (2) Apply matching property (quotient → subtraction), (3) Continue with inner structures (product → addition), (4) Apply power property last to bring all exponents down as coefficients. Work systematically from outside to inside, and always remember that exponents become coefficients in front!

Question 17

Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to write the expression as a single logarithm: $3\log_5(a)-\tfrac12\log_5(b)+\log_5(c).$

$\log_5\left(\dfrac{a^3c}{\sqrt{b}}\right)$ (correct answer)
$\log_5\left(\dfrac{a^3}{bc^2}\right)$
$\log_5\left(\dfrac{a^3c}{b}\right)$
$\log_5\left(a^3-\sqrt{b}+c\right)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you condense multiple logarithms into a single logarithm expression. The three logarithm properties come from exponent rules: (1) Product property: log_b(x) + log_b(y) = log_b(xy) because adding exponents means multiplying the original numbers, (2) Quotient property: log_b(x) - log_b(y) = log_b(x/y) because subtracting exponents means dividing, (3) Power property: p·log_b(x) = log_b(

x^p

) because a coefficient in front becomes an exponent inside. To condense 3log_5(a) - ½log_5(b) + log_5(c): First apply power property backwards to move coefficients up as exponents: log_5(a³) - log_5(b^(1/2)) + log_5(c). Note that b^(1/2) = √b. Next, work left to right combining terms: log_5(a³) - log_5(√b) becomes log_5(a³/√b) using quotient property. Finally add the last term: log_5(a³/√b) + log_5(c) becomes log_5((a³/√b)·c) = log_5(a³c/√b) using product property. Choice A correctly condenses the expression into a single logarithm with a³c in the numerator and √b in the denominator. Choice D incorrectly tries to combine logs using addition/subtraction of the arguments (a³ - √b + c), but there is NO logarithm property for log(x + y) or log(x - y)—you can only combine logs when they're being added (→ multiplication) or subtracted (→ division). Condensing strategy: (1) First convert all coefficients to exponents using power property backwards, (2) Then combine logs two at a time using product/quotient properties, (3) Addition of logs → multiplication inside, subtraction of logs → division inside. Remember: coefficients become exponents, and you can only combine logs through multiplication/division, never through addition/subtraction of arguments!

Question 18

$\ln\left(\dfrac{x^2y}{z^3}\right)$ (correct answer)
$\ln\left(\dfrac{x^2y}{3z}\right)$
$\ln\left(x^2+y- z^3\right)$
$\ln\left(\dfrac{x^2}{yz^3}\right)$

x^p

) moves coefficients up as exponents, (2) Product property: ln(x) + ln(y) = ln(xy) turns addition into multiplication, (3) Quotient property: ln(x) - ln(y) = ln(x/y) turns subtraction into division. To condense 2ln(x) + ln(y) - 3ln(z): First apply the power property to move coefficients up: ln(x²) + ln(y) - ln(z³). Next combine the first two terms using product property: ln(x²y) - ln(z³). Finally apply quotient property for the subtraction: ln(x²y/z³). Choice A correctly shows the condensed form as a single natural logarithm of the fraction x²y/z³. Choice B incorrectly interprets the coefficient 3 in front of ln(z) as going inside the log as 3z instead of as an exponent z³—the power property says 3ln(z) = ln(z³), not ln(3z)! This is a critical distinction: coefficients become exponents, not multipliers inside. Condensing strategy: (1) Convert all coefficients to exponents first using power property, (2) Identify additions (will become products) and subtractions (will become quotients), (3) Combine terms systematically, usually left to right, (4) The result should be a single log of one expression. Remember the pattern: coefficient → exponent, addition → multiplication, subtraction → division!

Question 19

Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to expand the expression $\log_3\left(\frac{x^4y}{z^2}\right)$ as a sum and/or difference of logarithms (no logarithms of products, quotients, or powers should remain).

$\log_3(x^4)+\log_3(y)-\log_3(z^2)$
$4\log_3(x)+\log_3(y)-2\log_3(z)$ (correct answer)
$\log_3(x)+\log_3(y)-\log_3(z)$
$\log_3\left(\frac{x^4+y}{z^2}\right)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you expand complex logarithmic expressions into sums and differences or condense multiple logs into a single logarithm. The three logarithm properties come from exponent rules and work because logarithms ARE exponents: (1) Product property: log_b(xy) = log_b(x) + log_b(y) because multiplying numbers means adding their exponents (b^m × b^n = b^(m+n)), (2) Quotient property: log_b(x/y) = log_b(x) - log_b(y) because dividing means subtracting exponents (b^m ÷ b^n = b^(m-n)), (3) Power property: log_b(

x^p

) = p·log_b(x) because raising to power means multiplying exponents ((b^m)^p = b^(mp)). To expand log_3(

x^4y/z^2

) using properties: (1) Apply quotient property first: log_3(

x^4y

) - log_3(

z^2

) because the overall structure is a fraction, (2) Apply product property to first log: log_3(

x^4

) + log_3(y) because x^4y is a product, (3) Apply power property to each: 4log_3(x) + log_3(y) - 2log_3(z) because x^4 has exponent 4 and z^2 has exponent 2. Choice B correctly applies all three properties in the right order to expand into sum and difference of simpler logs with coefficients. Choice A incorrectly leaves the powers inside the logarithms instead of bringing them out front as coefficients—the power property says log_b(

x^p

) = p·log_b(x), not log_b(

x^p

) by itself! Expanding logarithm strategy: (1) Identify overall structure: is it product, quotient, or power? Start with outermost operation, (2) Apply matching property: multiplication becomes +, division becomes -, exponent becomes coefficient, (3) Work from outside in: handle fraction first (quotient property), then products within (product property), then exponents (power property), (4) Continue until each log has single variable with coefficient. Remember: the goal of expanding is to have NO logarithms of products, quotients, or powers—just simple logs with coefficients in front!

Question 20

Expand the expression using logarithm properties (product, quotient, power):

$\log_3\left(\frac{x^2 y}{z^3}\right)$

Use:

$\log_b(xy)=\log_b(x)+\log_b(y)$
$\log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y)$
$\log_b(x^p)=p\log_b(x)$

$2\log_3(x)+\log_3(y)-3\log_3(z)$ (correct answer)
$\log_3(x+y)-\log_3(z^3)$
$\log_3(x^2)+\log_3(y)+\log_3(z^3)$
$\log_3(x)-\log_3(y)+3\log_3(z)$

Explanation: This question tests your understanding of the three fundamental logarithm properties—product, quotient, and power rules—that let you expand complex logarithmic expressions into sums and differences or condense multiple logs into a single logarithm. The three logarithm properties come from exponent rules and work because logarithms ARE exponents: (1) Product property: log_b(xy) = log_b(x) + log_b(y) because multiplying numbers means adding their exponents (b^m × b^n = b^(m+n)), (2) Quotient property: log_b(x/y) = log_b(x) - log_b(y) because dividing means subtracting exponents (b^m ÷ b^n = b^(m-n)), (3) Power property: log_b(

x^p

) = p·log_b(x) because raising to power means multiplying exponents ((b^m)^p = b^(mp)). To expand log_3(x²y/z³) using properties: (1) Apply quotient property first: log_3(x²y) - log_3(z³) because the overall structure is a fraction, (2) Apply product property to first log: log_3(x²) + log_3(y) because x²y is a product, (3) Apply power property to each: 2log_3(x) + log_3(y) - 3log_3(z) because x² has exponent 2 and z³ has exponent 3. Choice A correctly applies the logarithm properties in the right order to expand into sum and difference of simpler logs. Choice B incorrectly applies product property as log(x + y) = log(x) + log(y), but this is WRONG! The product property applies to multiplication: log(x·y) = log(x) + log(y), NOT to addition—there is NO property for log(x + y). Expanding logarithm strategy: (1) Identify overall structure: is it product, quotient, or power? Start with outermost operation, (2) Apply matching property: multiplication becomes +, division becomes -, exponent becomes coefficient, (3) Work from outside in: handle fraction first (quotient property), then products within (product property), then exponents (power property). Systematic outside-in approach ensures you correctly expand any complex logarithm into its simplest form!