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College Algebra Quiz

College Algebra Quiz: Log Properties Product Quotient Power

Practice Log Properties Product Quotient Power in College Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 20

0 of 20 answered

Given $\log_b(x)=2$ and $\log_b(y)=3$ , calculate $\log_b(x^2y)$ using properties.

What this quiz covers

This quiz focuses on Log Properties Product Quotient Power, giving you a quick way to practice the rules, question types, and explanations that matter most for College Algebra.

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

Given $\log_b(x)=2$ and $\log_b(y)=3$ , calculate $\log_b(x^2y)$ using properties.

$12$
$7$ (correct answer)
$5$
$9$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product, quotient, and power rules. Given log_b(x)=2 and log_b(y)=3, log_b(x^2 y) = 2*2 + 3 = 7. For this specific question, apply power to x^2 then product with y. Choice B is correct because 7 sums 4 and 3. Choice A is incorrect as 12 might multiply instead of add. To help students: Break into steps: power first, then product. Use to solve for actual values assuming a base.

Question 2

Which option correctly simplifies $\log_b\!\left(\frac{x}{y}\right)$ using the quotient rule?

$\log_b(x)-\log_b(y)$ (correct answer)
$\log_b(x)+\log_b(y)$
$\log_y(x)-\log_b(y)$
$\log_b\!\left(\frac{y}{x}\right)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the quotient rule. The quotient rule states that the logarithm of a quotient is the difference of the logarithms, written as (log_b(x/y) = log_b(x) - log_b(y)). For this specific question, the expression (log_b(x/y)) is simplified using the quotient rule. Choice A is correct because it properly subtracts the logarithms with the same base (b). Choice C is incorrect as it mixes bases and misapplies the rule, illustrating a frequent mistake in base consistency. To help students, work on problems that isolate the quotient rule before combining with others. Remind yourself to subtract the log of the denominator from the log of the numerator.

Question 3

How can $\log_b(x^n)$ be rewritten using the power rule?

$\log_b(x)\cdot n$
$n\,\log_b(x)$ (correct answer)
$\log_b(x)+\log_b(n)$
$\log_{10}(x^n)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the power rule. The power rule states that the logarithm of a power is the exponent times the logarithm, written as (log_b(x^n) = n log_b(x)). For this specific question, (log_b(x^n)) is rewritten using the power rule. Choice B is correct because it brings the exponent (n) in front of the logarithm correctly. Choice C is incorrect as it treats the exponent as a product instead of a power, showing a common confusion between rules. To help students, practice rewriting powers inside logs by pulling exponents out. Focus on distinguishing the power rule from the product rule in exercises.

Question 4

How can $\log_b\!\left(\sqrt{x}\right)$ be rewritten using the power rule?

$\frac{1}{2}\,\log_b(x)$ (correct answer)
$2\,\log_b(x)$
$\log_b(x)+\log_b(\tfrac{1}{2})$
$\log_b(2x)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the power rule. The power rule states that the logarithm of a power is the exponent times the logarithm, written as (log_b(x^n) = n log_b(x)). For this specific question, (log_b(sqrt{x}) = log_b(x^{1/2}) = rac{1}{2} log_b(x)). Choice A is correct because it uses the fractional exponent correctly. Choice B is incorrect as it uses 2 instead of 1/2. To help students, rewrite roots as fractional powers first. Practice with square roots and cube roots.

Question 5

Given $\log_b(x)=4$ and $\log_b(y)=1$ , calculate $\log_b(xy)$ .

$3$
$4$
$5$ (correct answer)
$1$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product rule. The product rule states that the logarithm of a product is the sum of the logarithms, written as (log_b(xy) = log_b(x) + log_b(y)). For this specific question, given (log_b(x) = 4) and (log_b(y) = 1), (log_b(xy) = 4 + 1 = 5). Choice C is correct because it adds to 5. Choice A is incorrect as 3 doesn't match the sum. To help students, add given values directly for products. Use to solve for unknown logs in equations.

Question 6

Which option correctly simplifies $\log_b\!\left(\frac{1}{x}\right)$ using the quotient rule?

$\log_b(1)-\log_b(x)$ (correct answer)
$\log_b(1)+\log_b(x)$
$\log_b(x)-\log_b(1)$
$\log_{10}(1)-\log_{10}(x)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the quotient rule. The quotient rule states that the logarithm of a quotient is the difference of the logarithms, written as (log_b(x/y) = log_b(x) - log_b(y)). For this specific question, (log_b(1/x)) uses the quotient rule with 1 as numerator. Choice A is correct because (log_b(1) - log_b(x)) is accurate, and (log_b(1) = 0). Choice B is incorrect as it adds, which is for products. To help students, recall that (log_b(1) = 0) always. Use this to simplify reciprocals to negative logs.

Question 7

Which option correctly simplifies $\log_b\!\left(\frac{xy}{z}\right)$ using the quotient rule?

$\log_b(xy)+\log_b(z)$
$\log_b(x)+\log_b(y)-\log_b(z)$ (correct answer)
$\log_b(x+y)-\log_b(z)$
$\log_z(xy)-\log_b(z)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the quotient and product rules. The quotient rule is (log_b(x/y) = log_b(x) - log_b(y)), and product expands sums. For this specific question, (log_b(xy/z) = log_b(x) + log_b(y) - log_b(z)). Choice B is correct because it combines product in numerator and quotient for denominator. Choice C is incorrect as it adds arguments, not logs. To help students, rewrite the fraction as a product with reciprocal. Apply rules in sequence for complex expressions.

Question 8

Which of the following correctly applies the product rule to $\log_b\!\left(\frac{x}{y}\cdot z\right)$ ?

$\log_b\!\left(\frac{x}{y}\right)+\log_b(z)$ (correct answer)
$\log_b\!\left(\frac{x}{y}+z\right)$
$\log_b\!\left(\frac{x}{y}\right)-\log_b(z)$
$\log_z\!\left(\frac{x}{y}\right)+\log_b(z)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product and quotient rules. The product rule adds logs, and quotient subtracts. For this specific question, (log_b( (x/y) z ) = log_b(x/y) + log_b(z)). Choice A is correct because it multiplies by z, adding its log. Choice C is incorrect as it subtracts z's log. To help students, rewrite the expression to group quotients and products. Apply rules step by step for mixed operations.

Question 9

If $\log_b(x)=m$ and $\log_b(y)=n$ , what is $\log_b\!\left(\frac{x}{y}\right)$ ?

$m+n$
$m-n$ (correct answer)
$mn$
$\log_b(x-y)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the quotient rule. The quotient rule states that the logarithm of a quotient is the difference of the logarithms, written as (log_b(x/y) = log_b(x) - log_b(y)). For this specific question, given (log_b(x) = m) and (log_b(y) = n), (log_b(x/y) = m - n). Choice B is correct because it subtracts the given values. Choice A is incorrect as it adds, confusing with product rule. To help students, use subtraction for division reminders. Substitute numbers to check calculations.

Question 10

Which of the following correctly applies the product rule to $\log_b(7x)$ ?

$\log_b(7+x)$
$\log_b(7)+\log_b(x)$ (correct answer)
$\log_7(7)+\log_b(x)$
$\log_{10}(7)+\log_{10}(x)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product rule. The product rule states that the logarithm of a product is the sum of the logarithms, written as (log_b(xy) = log_b(x) + log_b(y)). For this specific question, (log_b(7x)) is simplified using the product rule. Choice B is correct because it separates the constant 7 and (x) with the same base. Choice D is incorrect as it changes the base to 10, which is unnecessary and wrong. To help students, identify constants as logs of themselves. Practice with numerical constants to build intuition.

Question 11

Which option correctly simplifies $\log_b\!\left(\frac{xy}{y}\right)$ using the quotient rule?

$\log_b(xy)-\log_b(y)$ (correct answer)
$\log_b(xy)+\log_b(y)$
$\log_y(xy)-\log_b(y)$
$\log_b(x+y)-\log_b(y)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the quotient rule. The quotient rule states that the logarithm of a quotient is the difference of the logarithms, written as (log_b(x/y) = log_b(x) - log_b(y)). For this specific question, (log_b(xy / y) = log_b(xy) - log_b(y)), simplifying to (log_b(x)). Choice A is correct because it sets up the difference correctly. Choice B is incorrect as it adds, which wouldn't simplify properly. To help students, cancel terms algebraically before applying logs. See how the rule leads to cancellation in logs.

Question 12

If $\log_b(x)=m$ , what is $\log_b(x^5)$ using the power rule?

$m+5$
$5m$ (correct answer)
$m^5$
$\log_b(5x)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the power rule. The power rule states that the logarithm of a power is the exponent times the logarithm, written as (log_b(x^n) = n log_b(x)). For this specific question, given (log_b(x) = m), (log_b(x^5) = 5m). Choice B is correct because it multiplies by 5. Choice A is incorrect as it adds 5, not multiplies. To help students, think of the exponent as a coefficient. Use this to solve exponential equations.

Question 13

How can $\log_b\!\left((x^2)^5\right)$ be rewritten using the power rule?

$5\,\log_b(x^2)$ (correct answer)
$\log_b(x^{10})+5$
$\log_b(x^2)\cdot \log_b(5)$
$\log_b(x^2)+5$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the power rule. The power rule states that the logarithm of a power is the exponent times the logarithm, written as (log_b(x^n) = n log_b(x)). For this specific question, (log_b((x^2)^5) = 5 log_b(x^2)), which is equivalent to 10 log_b(x). Choice A is correct because it applies the power rule directly to the outer exponent. Choice B is incorrect as it adds 5 unnecessarily. To help students, simplify exponents inside first, like ( (x^2)^5 = x^{10} ). Then apply the rule to confirm.

Question 14

Which option correctly applies the product rule to $\log_b(xyz)$ ?

$\log_b(x)+\log_b(y)+\log_b(z)$ (correct answer)
$\log_b(x+y+z)$
$\log_b(x)+\log_c(y)+\log_b(z)$
$\log_{10}(xyz)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product rule. The product rule states that the logarithm of a product is the sum of the logarithms, written as (log_b(xy) = log_b(x) + log_b(y)), extending to more terms. For this specific question, (log_b(xyz)) expands to three logs added. Choice A is correct because it sums all three with base (b). Choice B is incorrect as it sums arguments, not logs. To help students, extend the rule to multiple factors iteratively. Practice with three or more variables.

Question 15

Given $\log_b(x)=2$ and $\log_b(y)=3$ , calculate $\log_b(x^2y)$ .

$7$ (correct answer)
$12$
$5$
$6$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product and power rules. The power rule is (log_b(x^n) = n log_b(x)), and product rule is (log_b(xy) = log_b(x) + log_b(y)). For this specific question, (log_b(x^2 y) = 2 log_b(x) + log_b(y) = 4 + 3 = 7). Choice A is correct because it computes to 7 using the rules. Choice B is incorrect as 12 would imply multiplication, not addition. To help students, break down combined expressions step by step. Use given values to plug in and calculate numerically.

Question 16

Which of the following correctly applies the product rule to $\log_b(ax)$ ?

$\log_b(a)+\log_b(x)$ (correct answer)
$\log_b(a+x)$
$\log_a(a)+\log_b(x)$
$\log_{10}(a)+\log_b(x)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product rule. The product rule states that the logarithm of a product is the sum of the logarithms, written as (log_b(xy) = log_b(x) + log_b(y)). For this specific question, (log_b(ax)) is simplified using the product rule. Choice A is correct because it adds the logs with the same base. Choice B is incorrect as it adds the arguments instead of the logs. To help students, treat variables like constants in products. Practice with different variables to generalize the rule.

Question 17

Given $\log_b(x)=1$ and $\log_b(y)=2$ , calculate $\log_b\!\left(\frac{x}{y}\right)$ .

$2$
$-1$ (correct answer)
$1$
$0$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the quotient rule. The quotient rule states that the logarithm of a quotient is the difference of the logarithms, written as (log_b(x/y) = log_b(x) - log_b(y)). For this specific question, given (log_b(x) = 1) and (log_b(y) = 2), (log_b(x/y) = 1 - 2 = -1). Choice B is correct because it computes to -1. Choice A is incorrect as 2 would be addition. To help students, handle negative results by remembering logs can be negative. Use to find logs of fractions.

Question 18

Which option correctly simplifies $\log_b\!\left(\frac{(xy)^2}{z}\right)$ ?

$2\,\log_b(x)+2\,\log_b(y)-\log_b(z)$ (correct answer)
$\log_b(x)+\log_b(y)-2\,\log_b(z)$
$\left(\log_b(x)+\log_b(y)\right)^2-\log_b(z)$
$2\,\log_b(xy-z)$

Explanation: This question tests college-level understanding of logarithmic properties, integrating power, product, and quotient rules. Power rule: log_b(a^n) = n log_b(a), applied to (xy)^2. For this specific question, log_b((xy)^2 / z) = 2 log_b(xy) - log_b(z) = 2 log_b(x) + 2 log_b(y) - log_b(z). Choice A is correct by expanding fully. Choice B incorrectly applies power to z. To help students: Expand grouped terms like (xy)^2 first. Verify by plugging in numbers for x, y, z.

Question 19

Given $\log_b(x)=m$ and $\log_b(y)=n$ , what is $\log_b(xy)$ ?

$mn$
$m-n$
$m+n$ (correct answer)
$\log_b(x+y)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product rule. The product rule states that the logarithm of a product is the sum of the logarithms, written as (log_b(xy) = log_b(x) + log_b(y)). For this specific question, given (log_b(x) = m) and (log_b(y) = n), (log_b(xy)) simplifies to (m + n). Choice C is correct because it adds the given logarithmic values as per the product rule. Choice B is incorrect as it subtracts instead of adding, confusing product with quotient. To help students, substitute numerical values to verify rules. Emphasize how given logs can be combined directly using properties.

Question 20

How can $\log_b\!\left(\left(\frac{x}{y}\right)^2\right)$ be rewritten using the power rule?

$\log_b\!\left(\frac{x}{y}\right)+2$
$2\,\log_b\!\left(\frac{x}{y}\right)$ (correct answer)
$\log_b(x^2)-\log_b(y)$
$\log_b(x)-\log_b(y^2)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the power rule. The power rule states that the logarithm of a power is the exponent times the logarithm, written as (log_b(x^n) = n log_b(x)). For this specific question, (log_b((x/y)^2)) is rewritten using the power rule. Choice B is correct because it multiplies the entire quotient's log by 2. Choice C is incorrect as it expands partially without the coefficient on both. To help students, apply power to the whole argument first. Then expand if needed for further simplification.

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College Algebra Quiz

College Algebra Quiz: Log Properties Product Quotient Power

Practice Log Properties Product Quotient Power in College Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 20

0 of 20 answered

Given $\log_b(x)=2$ and $\log_b(y)=3$ , calculate $\log_b(x^2y)$ using properties.

What this quiz covers

This quiz focuses on Log Properties Product Quotient Power, giving you a quick way to practice the rules, question types, and explanations that matter most for College Algebra.

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

Given $\log_b(x)=2$ and $\log_b(y)=3$ , calculate $\log_b(x^2y)$ using properties.

$12$
$7$ (correct answer)
$5$
$9$

Question 2

Which option correctly simplifies $\log_b\!\left(\frac{x}{y}\right)$ using the quotient rule?

$\log_b(x)-\log_b(y)$ (correct answer)
$\log_b(x)+\log_b(y)$
$\log_y(x)-\log_b(y)$
$\log_b\!\left(\frac{y}{x}\right)$

Question 3

How can $\log_b(x^n)$ be rewritten using the power rule?

$\log_b(x)\cdot n$
$n\,\log_b(x)$ (correct answer)
$\log_b(x)+\log_b(n)$
$\log_{10}(x^n)$

Question 4

How can $\log_b\!\left(\sqrt{x}\right)$ be rewritten using the power rule?

$\frac{1}{2}\,\log_b(x)$ (correct answer)
$2\,\log_b(x)$
$\log_b(x)+\log_b(\tfrac{1}{2})$
$\log_b(2x)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the power rule. The power rule states that the logarithm of a power is the exponent times the logarithm, written as (log_b(x^n) = n log_b(x)). For this specific question, (log_b(sqrt{x}) = log_b(x^{1/2}) = rac{1}{2} log_b(x)). Choice A is correct because it uses the fractional exponent correctly. Choice B is incorrect as it uses 2 instead of 1/2. To help students, rewrite roots as fractional powers first. Practice with square roots and cube roots.

Question 5

Given $\log_b(x)=4$ and $\log_b(y)=1$ , calculate $\log_b(xy)$ .

$3$
$4$
$5$ (correct answer)
$1$

Question 6

Which option correctly simplifies $\log_b\!\left(\frac{1}{x}\right)$ using the quotient rule?

$\log_b(1)-\log_b(x)$ (correct answer)
$\log_b(1)+\log_b(x)$
$\log_b(x)-\log_b(1)$
$\log_{10}(1)-\log_{10}(x)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the quotient rule. The quotient rule states that the logarithm of a quotient is the difference of the logarithms, written as (log_b(x/y) = log_b(x) - log_b(y)). For this specific question, (log_b(1/x)) uses the quotient rule with 1 as numerator. Choice A is correct because (log_b(1) - log_b(x)) is accurate, and (log_b(1) = 0). Choice B is incorrect as it adds, which is for products. To help students, recall that (log_b(1) = 0) always. Use this to simplify reciprocals to negative logs.

Question 7

Which option correctly simplifies $\log_b\!\left(\frac{xy}{z}\right)$ using the quotient rule?

$\log_b(xy)+\log_b(z)$
$\log_b(x)+\log_b(y)-\log_b(z)$ (correct answer)
$\log_b(x+y)-\log_b(z)$
$\log_z(xy)-\log_b(z)$

Question 8

Which of the following correctly applies the product rule to $\log_b\!\left(\frac{x}{y}\cdot z\right)$ ?

$\log_b\!\left(\frac{x}{y}\right)+\log_b(z)$ (correct answer)
$\log_b\!\left(\frac{x}{y}+z\right)$
$\log_b\!\left(\frac{x}{y}\right)-\log_b(z)$
$\log_z\!\left(\frac{x}{y}\right)+\log_b(z)$

Question 9

If $\log_b(x)=m$ and $\log_b(y)=n$ , what is $\log_b\!\left(\frac{x}{y}\right)$ ?

$m+n$
$m-n$ (correct answer)
$mn$
$\log_b(x-y)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the quotient rule. The quotient rule states that the logarithm of a quotient is the difference of the logarithms, written as (log_b(x/y) = log_b(x) - log_b(y)). For this specific question, given (log_b(x) = m) and (log_b(y) = n), (log_b(x/y) = m - n). Choice B is correct because it subtracts the given values. Choice A is incorrect as it adds, confusing with product rule. To help students, use subtraction for division reminders. Substitute numbers to check calculations.

Question 10

Which of the following correctly applies the product rule to $\log_b(7x)$ ?

$\log_b(7+x)$
$\log_b(7)+\log_b(x)$ (correct answer)
$\log_7(7)+\log_b(x)$
$\log_{10}(7)+\log_{10}(x)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product rule. The product rule states that the logarithm of a product is the sum of the logarithms, written as (log_b(xy) = log_b(x) + log_b(y)). For this specific question, (log_b(7x)) is simplified using the product rule. Choice B is correct because it separates the constant 7 and (x) with the same base. Choice D is incorrect as it changes the base to 10, which is unnecessary and wrong. To help students, identify constants as logs of themselves. Practice with numerical constants to build intuition.

Question 11

Which option correctly simplifies $\log_b\!\left(\frac{xy}{y}\right)$ using the quotient rule?

$\log_b(xy)-\log_b(y)$ (correct answer)
$\log_b(xy)+\log_b(y)$
$\log_y(xy)-\log_b(y)$
$\log_b(x+y)-\log_b(y)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the quotient rule. The quotient rule states that the logarithm of a quotient is the difference of the logarithms, written as (log_b(x/y) = log_b(x) - log_b(y)). For this specific question, (log_b(xy / y) = log_b(xy) - log_b(y)), simplifying to (log_b(x)). Choice A is correct because it sets up the difference correctly. Choice B is incorrect as it adds, which wouldn't simplify properly. To help students, cancel terms algebraically before applying logs. See how the rule leads to cancellation in logs.

Question 12

If $\log_b(x)=m$ , what is $\log_b(x^5)$ using the power rule?

$m+5$
$5m$ (correct answer)
$m^5$
$\log_b(5x)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the power rule. The power rule states that the logarithm of a power is the exponent times the logarithm, written as (log_b(x^n) = n log_b(x)). For this specific question, given (log_b(x) = m), (log_b(x^5) = 5m). Choice B is correct because it multiplies by 5. Choice A is incorrect as it adds 5, not multiplies. To help students, think of the exponent as a coefficient. Use this to solve exponential equations.

Question 13

How can $\log_b\!\left((x^2)^5\right)$ be rewritten using the power rule?

$5\,\log_b(x^2)$ (correct answer)
$\log_b(x^{10})+5$
$\log_b(x^2)\cdot \log_b(5)$
$\log_b(x^2)+5$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the power rule. The power rule states that the logarithm of a power is the exponent times the logarithm, written as (log_b(x^n) = n log_b(x)). For this specific question, (log_b((x^2)^5) = 5 log_b(x^2)), which is equivalent to 10 log_b(x). Choice A is correct because it applies the power rule directly to the outer exponent. Choice B is incorrect as it adds 5 unnecessarily. To help students, simplify exponents inside first, like ( (x^2)^5 = x^{10} ). Then apply the rule to confirm.

Question 14

Which option correctly applies the product rule to $\log_b(xyz)$ ?

$\log_b(x)+\log_b(y)+\log_b(z)$ (correct answer)
$\log_b(x+y+z)$
$\log_b(x)+\log_c(y)+\log_b(z)$
$\log_{10}(xyz)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product rule. The product rule states that the logarithm of a product is the sum of the logarithms, written as (log_b(xy) = log_b(x) + log_b(y)), extending to more terms. For this specific question, (log_b(xyz)) expands to three logs added. Choice A is correct because it sums all three with base (b). Choice B is incorrect as it sums arguments, not logs. To help students, extend the rule to multiple factors iteratively. Practice with three or more variables.

Question 15

Given $\log_b(x)=2$ and $\log_b(y)=3$ , calculate $\log_b(x^2y)$ .

$7$ (correct answer)
$12$
$5$
$6$

Question 16

Which of the following correctly applies the product rule to $\log_b(ax)$ ?

$\log_b(a)+\log_b(x)$ (correct answer)
$\log_b(a+x)$
$\log_a(a)+\log_b(x)$
$\log_{10}(a)+\log_b(x)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product rule. The product rule states that the logarithm of a product is the sum of the logarithms, written as (log_b(xy) = log_b(x) + log_b(y)). For this specific question, (log_b(ax)) is simplified using the product rule. Choice A is correct because it adds the logs with the same base. Choice B is incorrect as it adds the arguments instead of the logs. To help students, treat variables like constants in products. Practice with different variables to generalize the rule.

Question 17

Given $\log_b(x)=1$ and $\log_b(y)=2$ , calculate $\log_b\!\left(\frac{x}{y}\right)$ .

$2$
$-1$ (correct answer)
$1$
$0$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the quotient rule. The quotient rule states that the logarithm of a quotient is the difference of the logarithms, written as (log_b(x/y) = log_b(x) - log_b(y)). For this specific question, given (log_b(x) = 1) and (log_b(y) = 2), (log_b(x/y) = 1 - 2 = -1). Choice B is correct because it computes to -1. Choice A is incorrect as 2 would be addition. To help students, handle negative results by remembering logs can be negative. Use to find logs of fractions.

Question 18

Which option correctly simplifies $\log_b\!\left(\frac{(xy)^2}{z}\right)$ ?

$2\,\log_b(x)+2\,\log_b(y)-\log_b(z)$ (correct answer)
$\log_b(x)+\log_b(y)-2\,\log_b(z)$
$\left(\log_b(x)+\log_b(y)\right)^2-\log_b(z)$
$2\,\log_b(xy-z)$

Question 19

Given $\log_b(x)=m$ and $\log_b(y)=n$ , what is $\log_b(xy)$ ?

$mn$
$m-n$
$m+n$ (correct answer)
$\log_b(x+y)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the product rule. The product rule states that the logarithm of a product is the sum of the logarithms, written as (log_b(xy) = log_b(x) + log_b(y)). For this specific question, given (log_b(x) = m) and (log_b(y) = n), (log_b(xy)) simplifies to (m + n). Choice C is correct because it adds the given logarithmic values as per the product rule. Choice B is incorrect as it subtracts instead of adding, confusing product with quotient. To help students, substitute numerical values to verify rules. Emphasize how given logs can be combined directly using properties.

Question 20

How can $\log_b\!\left(\left(\frac{x}{y}\right)^2\right)$ be rewritten using the power rule?

$\log_b\!\left(\frac{x}{y}\right)+2$
$2\,\log_b\!\left(\frac{x}{y}\right)$ (correct answer)
$\log_b(x^2)-\log_b(y)$
$\log_b(x)-\log_b(y^2)$

Explanation: This question tests college-level understanding of logarithmic properties, specifically the power rule. The power rule states that the logarithm of a power is the exponent times the logarithm, written as (log_b(x^n) = n log_b(x)). For this specific question, (log_b((x/y)^2)) is rewritten using the power rule. Choice B is correct because it multiplies the entire quotient's log by 2. Choice C is incorrect as it expands partially without the coefficient on both. To help students, apply power to the whole argument first. Then expand if needed for further simplification.