📝 Logarithm properties product quotient power (18 MCQs)

📖 From Calculus • 1. Basics before calculus • 18 questions available

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What is Logarithm properties product quotient power?

Definition:
Logarithm properties include: $\log_b(MN) = \log_b(M) + \log_b(N)$ (product), $\log_b(M/N) = \log_b(M) - \log_b(N)$ (quotient), and $\log_b(M^p) = p \log_b(M)$ (power), which facilitate simplification and solving logarithmic equations.

Example:
$\log_2(8 \cdot 4) = \log_2(8) + \log_2(4) = 3+2=5$ ; $\log_5(25^3) = 3 \log_5(25) = 3 \cdot 2 = 6$ .

Reason:
These properties are fundamental for algebraic manipulation, making complex logarithmic expressions tractable in science and engineering.

Easy

Medium

Hard

📝 All Logarithm properties product quotient power MCQs

Q1. If $\log_{b}(xy)=5$ and $\log_{b}x=2$ , what is $\log_{b}y$ ?

A.2

B.3 ✅

C.5

D.7

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Apply the product property: $\log_{b}(xy)=\log_{b}x+\log_{b}y$ . Substituting gives $5=2+\log_{b}y$ , so $\log_{b}y=3$ . The correct choice is the option with value 3, which is B.

Q2. Which property correctly simplifies $\log_{a}(x^{3}y^{2})$ ?

\log_{a}x^{3} + \log_{a}y^{2}

3\log_{a}x + 2\log_{a}y

✅

\log_{a}(x^{3}) \cdot \log_{a}(y^{2})

\log_{a}x + \log_{a}y

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: First use the product property: $\log_{a}(x^{3}y^{2})=\log_{a}x^{3}+\log_{a}y^{2}$ . Then apply the power property to each term, giving $3\log_{a}x+2\log_{a}y$ . This matches option B.

Q3. Simplify $\log_{2}8 + 3\log_{2}x - \frac12\log_{2}x^{2}$ to a single logarithm.

\log_{2}(8x^{2})

✅

\log_{2}(x^{5})

\log_{2}(8x^{3})

\log_{2}(x^{2})

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Convert each term: $\log_{2}8=3$ . The half‑log term becomes $\frac12\log_{2}x^{2}= \log_{2}x$ . The expression is $3+3\log_{2}x-\log_{2}x = 3+2\log_{2}x$ . Using the product property gives $\log_{2}(2^{3}x^{2})=\log_{2}(8x^{2})$ , which is option A.

Q4. Given $\log_{10}(x^{2}) = 4$ , what is the value of $x$ ?

A.10

B.100 ✅

C.1000

D.-100

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Rewrite as an exponential: $x^{2}=10^{4}=10000$ . Since the argument of a logarithm must be positive, $x>0$ , so $x=100$ . The correct numerical value appears in option B.

Q5. Which equation represents the product property of logarithms?

\log_{b}(ac)=\log_{b}a+\log_{b}c

✅

\log_{b}(a/c)=\log_{b}a-\log_{b}c

\log_{b}(a^{r})=r\log_{b}a

\log_{b}(1/c)=-\log_{b}c

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The product property states that the logarithm of a product equals the sum of the logarithms of the factors: $\log_{b}(ac)=\log_{b}a+\log_{b}c$ . This matches option A, making it the correct choice.

Q6. Using the reciprocal property, simplify $\log_{5}\left(\frac{1}{x^{3}}\right)$ .

3\log_{5}x

-3\log_{5}x

✅

\log_{5}x^{3}

-\log_{5}x

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Apply the reciprocal property: $\log_{5}(1/x^{3})=-\log_{5}(x^{3})$ . Then use the power property on the right‑hand side, giving $-3\log_{5}x$ . This corresponds to option B.

Q7. If $\log_{a}b = 2$ and $\log_{b}c = 3$ , what is $\log_{a}c$ ?

A.5

B.6 ✅

C.8

D.9

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: By the change‑of‑base formula, $\log_{a}c = \log_{a}b \cdot \log_{b}c = 2 \times 3 = 6$ . Therefore the correct answer is 6, which is option B.

Q8. Identify the error in the simplification $\log (x + y) = \log x + \log y$ .

A.Misuse of the product property ✅

B.Misuse of the power property

C.Correct application of the quotient property

D.No error; the equality holds

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The product property applies only to multiplication, not addition. Treating $x+y$ as a product incorrectly leads to $\log (x + y) = \log x + \log y$ . Thus the mistake is a misuse of the product property, corresponding to option A.

Q9. Solve for $x$ : $2\log_{3}x - \log_{3}(x-2) = 1$ .

x = 3

x = 6

C.No real solution ✅

x = 2

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Combine logs: $\log_{3}(x^{2}) - \log_{3}(x-2)=\log_{3}\!\left(\frac{x^{2}}{x-2}\right)=1$ . Exponentiate to get $\frac{x^{2}}{x-2}=3$ → $x^{2}=3x-6$ → $x^{2}-3x+6=0$ . The discriminant is negative, so no real roots exist, making option C correct.

Q10. What is the power property of logarithms?

\log_{b}(ac)=\log_{b}a+\log_{b}c

\log_{b}(a/c)=\log_{b}a-\log_{b}c

\log_{b}(a^{r})=r\log_{b}a

✅

\log_{b}(1/c)=-\log_{b}c

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The power property states that the logarithm of a power equals the exponent multiplied by the logarithm of the base: $\log_{b}(a^{r}) = r\log_{b}a$ . This description matches option C.

Q11. If $\log_{2}8 = a$ and $\log_{2}4 = b$ , what is $a - b$ ?

A.0

B.1 ✅

C.2

D.3

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Compute the values: $\log_{2}8 = 3$ and $\log_{2}4 = 2$ . Their difference is $3-2 = 1$ . Hence $a-b = 1$ , which corresponds to option B.

Q12. Which statement correctly relates $\log_{b}(a^{r})$ and $r\log_{b}a$ ?

A.They are always equal ✅

B.They are equal only when

r

is an integer

C.They are never equal

D.Equality holds only for

b=10

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The power property guarantees that for any real exponent $r$ , $\log_{b}(a^{r}) = r\log_{b}a$ . Thus the two expressions are identically equal for all permissible values, making option A the correct statement.

Q13. Simplify $\log_{10}\left(\frac{1000}{10}\right)$ .

A.2 ✅

B.3

C.1

D.0

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: $\frac{1000}{10}=100$ . Using the quotient property, $\log_{10}100 = \log_{10}10^{2}=2$ . Therefore the simplified value is 2, which is option A.

Q14. Express $\log_{10}\left(\frac{x^{2}y^{3}}{\sqrt{y}}\right)$ in terms of $p=\log_{10}x$ and $q=\log_{10}y$ .

2p + \frac{5}{2}q

✅

2p + \frac{3}{2}q

p + 3q

2p + 2q

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Break the expression: $\log_{10}x^{2}=2p$ , $\log_{10}y^{3}=3q$ , $\log_{10}\sqrt{y}= \frac12 q$ . Apply the quotient property: $2p + 3q - \frac12 q = 2p + \frac{5}{2}q$ . This matches option A.

Q15. Find the base $b$ such that $\log_{b}27 = \frac{3}{2}$ .

A.3

B.6

C.9 ✅

D.12

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Rewrite as $b^{3/2}=27$ . Raise both sides to the $2/3$ power: $b = 27^{2/3} = (3^{3})^{2/3}=3^{2}=9$ . Hence the base is 9, which is option C.

Q16. Which property allows the conversion $\log_{b}\left(\frac{1}{c}\right) = -\log_{b}c$ ?

A.Product property

B.Quotient property

C.Power property

D.Reciprocal property ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The reciprocal property explicitly states that the logarithm of a reciprocal equals the negative of the logarithm of the original number: $\log_{b}(1/c) = -\log_{b}c$ . This is described by option D.

Q17. Which expression cannot be simplified using the standard logarithmic properties?

\log_{2}(x^{4}y)

\log_{3}\left(\frac{m}{n^{2}}\right)

\log_{5}(p+q)

✅

\log_{7}(a^{3})

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Logarithmic properties handle products, quotients, and powers, but not sums or differences inside a log. $\log_{5}(p+q)$ involves a sum, so it cannot be broken down further, making option C the correct choice.

Q18. If $\log_{5}x = 2$ , what is $\log_{x}5$ ?

\frac12

✅

B.2

C.4

\frac14

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: From $\log_{5}x = 2$ we get $x = 5^{2}=25$ . Using the change‑of‑base formula, $\log_{x}5 = 1/\log_{5}x = 1/2 = \frac12$ . Thus the correct answer is option A.)

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