ACCUPLACER Advanced Algebra & Functions Quiz: Logarithm Properties

8 questions · exam conditions

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Logarithm PropertiesQuestion 1 of 8

If $\log_2 x + \log_2 y = 5$ and $\log_2 x - \log_2 y = 1$ , what is the value of $xy$ ?

128

ACCUPLACER Advanced Algebra & Functions Quiz

ACCUPLACER Advanced Algebra & Functions Quiz: Logarithm Properties

Practice Logarithm Properties in ACCUPLACER Advanced Algebra & Functions 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 Logarithm Properties, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Advanced Algebra & Functions.

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

If $\log_2 x + \log_2 y = 5$ and $\log_2 x - \log_2 y = 1$ , what is the value of $xy$ ?

16
32 (correct answer)
64
128

Explanation: Using the product and quotient properties:

\log_2 x + \log_2 y = \log_2(xy) = 5

, so

xy = 2^5 = 32

. We can verify by solving the system: adding the equations gives

2\log_2 x = 6

, so

x = 8

. Subtracting gives

2\log_2 y = 4

, so

y = 4

. Thus

xy = 32

Question 2

For positive values of $x$ , $y$ , and $z$ , which expression is equivalent to $\log\left(\frac{x^3y^2}{z^4}\right)$ ?

$\log(3x) + \log(2y) - \log(4z)$
$\frac{3\log x + 2\log y}{4\log z}$
$3\log x + 2\log y - 4\log z$ (correct answer)
$\log(3x^2y) - \log(4z)$

Explanation: When you encounter logarithms of complex expressions involving multiplication, division, and exponents, you need to apply the fundamental logarithm properties to break them down into simpler components. The expression

\log\left(\frac{x^3y^2}{z^4}\right)

can be simplified using three key logarithm rules:

$\log\left(\frac{a}{b}\right) = \log a - \log b$ (quotient rule)
$\log(ab) = \log a + \log b$ (product rule)
$\log(a^n) = n\log a$ (power rule)

First, apply the quotient rule:

\log\left(\frac{x^3y^2}{z^4}\right) = \log(x^3y^2) - \log(z^4)

Next, use the product rule on the first term:

\log(x^3y^2) = \log(x^3) + \log(y^2)

Finally, apply the power rule to each term:

\log(x^3) = 3\log x

\log(y^2) = 2\log y

, and

\log(z^4) = 4\log z

Combining everything:

3\log x + 2\log y - 4\log z

, which is choice C. Choice A incorrectly brings the exponents inside the logarithms as coefficients of the variables. Choice B mistakenly treats the division as a fraction of logarithms rather than a difference. Choice D fails to properly separate all terms and incorrectly handles the exponents in both the numerator and denominator. Remember: exponents become coefficients when you break down logarithms, multiplication becomes addition, and division becomes subtraction. Always work systematically through the logarithm properties rather than trying to manipulate the expression intuitively.

Question 3

Which expression is equivalent to $\log_3(27x^2) - \log_3(9x)$ for $x > 0$ ?

$\log_3(3x)$ (correct answer)
$\log_3(18x)$
$\log_3(3) + \log_3(x)$
$\log_3(243x^3)$

Explanation: Using the quotient property:

\log_3(27x^2) - \log_3(9x) = \log_3\left(\frac{27x^2}{9x}\right) = \log_3(3x)

. Choice B incorrectly adds the arguments. Choice C equals

\log_3(3x)

but isn't the simplified form requested. Choice D incorrectly multiplies the arguments.

Question 4

Which of the following statements about logarithms is always true for positive real numbers $a$ , $b$ , and valid bases?

$\log(a + b) = \log a + \log b$
$\log(ab) = (\log a)(\log b)$
$\log\left(\frac{a}{b}\right) = \frac{\log a}{\log b}$
$\log(a^b) = b \log a$ (correct answer)

Explanation: The power property states

\log(a^b) = b \log a

is always true. Choice A is false;

\log(a + b) \neq \log a + \log b

. Choice B is false; the correct product property is

\log(ab) = \log a + \log b

. Choice C is false; the correct quotient property is

\log\left(\frac{a}{b}\right) = \log a - \log b

Question 5

Which expression represents the complete simplification of $\log_2(8x^3) + \log_2\left(\frac{2}{x}\right) - \log_2(4x^2)$ ?

$\log_2(x)$
$\log_2(1) = 0$
$\log_2(4) = 2$ (correct answer)
$\log_2(8) = 3$

Explanation: Combining using logarithm properties:

\log_2(8x^3) + \log_2\left(\frac{2}{x}\right) - \log_2(4x^2) = \log_2\left(\frac{8x^3 \cdot \frac{2}{x}}{4x^2}\right) = \log_2\left(\frac{16x^2}{4x^2}\right) = \log_2(4) = 2

Question 6

If $\log_5 a = 2$ and $\log_5 b = 3$ , what is the value of $\log_5\left(\frac{a^3}{b^2}\right)$ ?

$-3$
$0$ (correct answer)
$1$
$6$

Explanation: Using logarithm properties:

\log_5\left(\frac{a^3}{b^2}\right) = \log_5(a^3) - \log_5(b^2) = 3\log_5 a - 2\log_5 b = 3(2) - 2(3) = 6 - 6 = 0

. Choice A results from incorrectly calculating

2(3) - 3(2)

. Choice C comes from

3 - 2

. Choice D comes from

3(2)

Question 7

Given that $\log_3 2 \approx 0.631$ , which value is closest to $\log_3 72$ ?

$3.5$
$5.2$
$4.6$
$4.0$ (correct answer)

Explanation: When you encounter logarithm problems with unfamiliar bases, the key strategy is to break down complex expressions using logarithm properties, especially the product rule:

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

. To find

\log_3 72

, start by factoring 72 into familiar components. Since

72 = 8 \times 9

, we can write:

\log_3 72 = \log_3(8 \times 9) = \log_3 8 + \log_3 9

Now evaluate each piece. Since

9 = 3^2

, we know

\log_3 9 = 2

. For

\log_3 8

, notice that

8 = 2^3

, so:

\log_3 8 = \log_3(2^3) = 3\log_3 2 = 3(0.631) = 1.893

Therefore:

\log_3 72 = 1.893 + 2 = 3.893 \approx 4.0

This confirms answer choice D is correct. Choice A (3.5) likely comes from incorrectly calculating

\log_3 8

or making arithmetic errors in the final sum. Choice B (5.2) suggests confusion about logarithm properties, possibly adding when you should multiply or using the wrong base conversion. Choice C (4.6) might result from miscalculating

3 \times 0.631

or incorrectly handling the factorization of 72. Study tip: When working with unfamiliar logarithms, always factor the argument into powers of the base (like

3^2 = 9

) and powers of numbers you're given information about (like

2^3 = 8

). This makes complex logarithms much more manageable using basic logarithm properties.

Question 8

If $\log_a x = p$ and $\log_a y = q$ , then $\log_a\left(\frac{x^2\sqrt{y}}{a}\right)$ equals:

$2p + \frac{q}{2} - 1$ (correct answer)
$2p + \frac{q}{2} + 1$
$\frac{2pq}{2} - 1$
$2p\sqrt{q} - 1$

Explanation: Using logarithm properties:

\log_a\left(\frac{x^2\sqrt{y}}{a}\right) = \log_a(x^2\sqrt{y}) - \log_a(a) = \log_a(x^2) + \log_a(\sqrt{y}) - 1 = 2\log_a x + \frac{1}{2}\log_a y - 1 = 2p + \frac{q}{2} - 1

. Choice B incorrectly adds 1 instead of subtracting. Choice C incorrectly combines the first two terms. Choice D incorrectly takes the square root of q outside the logarithm.