📝 Change of base formula for logs (14 MCQs)

📖 From Calculus • 1. Basics before calculus • 14 questions available

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What is Change of base formula for logs?

Definition:
The change of base formula states that $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$ for any positive base $k \neq 1$ , allowing conversion between different logarithmic bases, commonly used to evaluate logs on calculators (base 10 or e).

Example:
To compute $\log_2(5)$ , use $\log_2(5) = \frac{\ln(5)}{\ln(2)} \approx \frac{1.6094}{0.6931} \approx 2.3219$ .

Reason:
This formula is essential when only certain bases are available in tools, and for comparing logs across different bases in theoretical work.

Easy

Medium

Hard

📝 All Change of base formula for logs MCQs

Q1. What is the change of base formula for logarithms?

A.\

\\log_{b} x = \\frac{\\ln x}{\\ln b} \

✅

B.\

\\log_{b} x = \\ln (x b) \

C.\

\\log_{b} x = \\frac{\\log_{10} x}{\\log_{10} b} \

D.\

\\log_{b} x = \\frac{b}{x} \

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The change of base formula follows from the definition $b^{y}=x$ . Taking natural logs gives $y\\ln b=\\ln x$ , so $y=\\frac{\\ln x}{\\ln b}$ . This identity holds for any admissible bases, making option A the correct representation.

Q2. Given that \ $\\log_{2} 8 = 3\$ , which expression correctly represents \ $\\log_{10} 8\$ using the change of base formula?

A.3 \\log_{10} 2

B.\\frac{3}{\\log_{2} 10} ✅

C.\\log_{10} 2 + 3

D.\\frac{\\log_{10} 2}{3}

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Applying the change of base formula, \ $\\log_{10} 8 = \\frac{\\log_{2} 8}{\\log_{2} 10}\$ . Since \ $\\log_{2} 8 = 3\$ , the expression simplifies to \ $\\frac{3}{\\log_{2} 10}\$ , which matches option B.

Q3. Which statement about the numerical relationship between \ $\\log_{2} 5\$ and \ $\\ln 5\$ is true?

A.\

\\log_{2} 5 = \\ln 5\

B.\

\\log_{2} 5 < \\ln 5\

C.\

\\log_{2} 5 > \\ln 5\

✅

D.Cannot be determined without a calculator

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Using the change of base formula, \ $\\log_{2} 5 = \\frac{\\ln 5}{\\ln 2}\$ . Because \ $\\ln 2 \\approx 0.693\$ is less than 1, dividing \ $\\ln 5\$ by a number smaller than 1 yields a larger result, so \ $\\log_{2} 5\$ is greater than \ $\\ln 5\$ . Option C states this correctly.

Q4. Using natural logarithms, evaluate \ $\\log_{5} 2\$ . Choose the closest approximation.

A.0.23

B.1.43

C.2.32

D.0.43 ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Apply the formula \ $\\log_{5} 2 = \\frac{\\ln 2}{\\ln 5}\$ . With \ $\\ln 2 \\approx 0.6931\$ and \ $\\ln 5 \\approx 1.6094\$ , the quotient is about 0.4307, which rounds to 0.43. Hence option D is the best approximation.

Q5. If \ $\\log_{a} b = 2\$ and \ $\\log_{b} c = 3\$ , what is \ $\\log_{a} c\$ expressed using the change of base concept?

A.5

B.6 ✅

C.01-Jun

D.3

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Write \ $\\log_{a} c = \\log_{a} b \\cdot \\log_{b} c\$ . Substituting the given values gives \ $2 \\times 3 = 6\$ . This product follows from the property \ $\\log_{a} c = \\frac{\\ln c}{\\ln a}\$ and the intermediate base b, confirming option B.

Q6. Which of the following expressions is equivalent to \ $\\log_{4} 8\$ by applying the change of base formula with natural logs?

A.\\frac{\\ln 8}{\\ln 4} ✅

B.\\frac{\\ln 4}{\\ln 8}

C.\\ln(8-4)

D.\\ln 8 \\cdot \\ln 4

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The change of base formula states \ $\\log_{4} 8 = \\frac{\\ln 8}{\\ln 4}\$ . This directly matches option A, while the other options either invert the ratio or perform unrelated operations.

Q7. Why does the change of base formula work for any positive bases b and c (b,c \\neq 1)?

A.Because taking natural logs of the equation \

b^{y}=x\

yields \

y = \\frac{\\ln x}{\\ln b}\

✅

B.It is a coincidence that only holds for base 10

C.It requires b to be an integer

D.It only works for rational bases

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Starting from the definition $b^{y}=x$ and applying the natural logarithm gives $y\\ln b = \\ln x$ . Solving for y produces $y = \\frac{\\ln x}{\\ln b}$ , which is valid for any admissible bases, establishing the universal applicability described in option A.

Q8. Given \ $\\ln 7 \\approx 1.9459\$ and \ $\\ln 2 \\approx 0.6931\$ , estimate \ $\\log_{2} 7\$ using the change of base formula.

A.2.8 ✅

B.3

C.2

D.1.8

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Using the formula \ $\\log_{2} 7 = \\frac{\\ln 7}{\\ln 2}\$ , divide 1.9459 by 0.6931 to obtain approximately 2.805, which rounds to 2.80. This matches option A, while the other choices differ noticeably.

Q9. Among the bases b = 2, 3, 10, and e, which yields the smallest (most negative) value for \ $\\log_{b} 0.5\$ ?

A.b = 2 ✅

B.b = 3

C.b = 10

D.b = e

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Compute each: \ $\\log_{2} 0.5 = -1\$ , \ $\\log_{3} 0.5 \\approx -0.631\$ , \ $\\log_{10} 0.5 \\approx -0.301\$ , \ $\\log_{e} 0.5 \\approx -0.693\$ . The most negative is -1, occurring for base 2, so option A is correct.

Q10. Express \ $\\log_{12} 18\$ in terms of common (base‑10) logarithms.

A.\\frac{\\log_{10} 18}{\\log_{10} 12} ✅

B.\\frac{\\log_{10} 12}{\\log_{10} 18}

C.\\log_{10}(18-12)

D.\\log_{10} 18 \\times \\log_{10} 12

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Applying the change of base formula with base 10 gives \ $\\log_{12} 18 = \\frac{\\log_{10} 18}{\\log_{10} 12}\$ . This direct substitution matches option A, whereas the other options either invert the ratio or perform unrelated operations.

Q11. Prove that \ $\\log_{b} x = \\frac{1}{\\log_{x} b}\$ using the change of base formula.

A.Apply the formula \

\\log_{b} x = \\frac{\\ln x}{\\ln b}\

and invert it ✅

B.Use exponent rules directly

C.Assume the result as a definition

D.The statement is false

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: From the change of base formula, \ $\\log_{b} x = \\frac{\\ln x}{\\ln b}\$ . Taking the reciprocal gives \ $\\frac{1}{\\log_{b} x} = \\frac{\\ln b}{\\ln x} = \\log_{x} b\$ . Rearranging yields \ $\\log_{b} x = \\frac{1}{\\log_{x} b}\$ , confirming option A.

Q12. Given \ $\\log_{10} 2 = 0.3010\$ and \ $\\log_{10} 3 = 0.4771\$ , compute \ $\\log_{2} 9\$ without a calculator.

A.3.17 ✅

B.2

C.4

D.1.5

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: First find \ $\\log_{10} 9 = \\log_{10}(3^2) = 2\\times0.4771 = 0.9542\$ . Then apply change of base: \ $\\log_{2} 9 = \\frac{\\log_{10} 9}{\\log_{10} 2} = \\frac{0.9542}{0.3010} \\approx 3.17\$ . Option A provides this value.

Q13. When the base of a logarithmic function is changed from b to e, how does the graph of \ $f(x)=\\log_{b} x\$ transform?

A.It is vertically stretched by a factor of \

1/\\ln b\

while the vertical asymptote at x=0 remains unchanged ✅

B.It shifts right by \

\\ln b\

C.It becomes periodic

D.There is no change in shape

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Using the formula \ $\\log_{b} x = \\frac{\\ln x}{\\ln b}\$ , the function is a scaled version of \ $\\ln x\$ . The scaling factor is \ $1/\\ln b\$ , which stretches or compresses the graph vertically but does not move the asymptote at x=0. Hence option A describes the transformation accurately.

Q14. An algorithm computes \ $\\log_{b} x\$ by first evaluating \ $\\ln x\$ and \ $\\ln b\$ then applying the change of base formula. If evaluating a natural log takes O(n) time, what is the overall time complexity?

A.O(n)

B.O(n^2)

C.O(n) plus a constant amount of extra work ✅

D.O(log n)

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The algorithm performs two independent natural‑log evaluations, each costing O(n). After those calls, only a constant‑time division and multiplication are needed. Adding the two O(n) operations yields overall O(n) work, plus a constant overhead, which is described by option C.

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