📝 Derivatives of Logarithmic Functions (11 MCQs)

📖 From Calculus • 4. Topics in Differentiation • 11 questions available

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What is Derivatives of Logarithmic Functions?

Definition:
The derivative of a logarithmic function measures the rate of change of the logarithm with respect to its argument. For the natural logarithm $\ln(x)$ , the derivative is $\frac{1}{x}$ . For a general base $b$ , the derivative of $\log_b(x)$ involves dividing by the natural logarithm of the base, reflecting how the scaling factor of the logarithm affects its slope at any given positive input value.

Example:
Find the derivative of $f(x) = \log_2(x)$ . Using the formula $\frac{d}{dx}\log_b(x) = \frac{1}{x \ln(b)}$ , we get $f'(x) = \frac{1}{x \ln(2)}$ . If $x=1$ , the slope is $\frac{1}{\ln(2)} \approx 1.44$ .

Reason:
These derivatives are fundamental in modeling phenomena involving exponential decay or growth, such as pH levels in chemistry or sound intensity in decibels, where logarithmic scales are used to compress wide ranges of values.

Easy

Medium

Hard

📝 All Derivatives of Logarithmic Functions MCQs

Q1. What is the derivative of $\ln x$ for $x>0$ ?

\frac{1}{x}

✅

x

\ln x

e^{x}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Using the definition of derivative for the natural logarithm, one obtains $\frac{d}{dx}\ln x = \lim_{h\to0}\frac{\ln(x+h)-\ln x}{h}= \frac{1}{x}$ for all $x>0$ . Hence option A is correct.

Q2. Given $f(x)=\log_{2}(x^{3})$ , which expression correctly represents $f'(x)$ ?

\frac{1}{3x\ln 2}

\frac{3\ln 2}{x}

\frac{3}{x\ln 2}

✅

\frac{1}{x\ln 2^{3}}

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Rewrite $f(x)=\log_{2}(x^{3}) = \frac{\ln(x^{3})}{\ln 2}= \frac{3\ln x}{\ln 2}$ . Differentiating gives $f'(x)=\frac{3}{x\ln 2}$ . This matches option C, confirming the logical deduction from the properties of logarithms and the constant‑multiple rule.

Q3. How does the derivative of $\log_{10} x$ compare to the derivative of $\ln x$ ?

A.Both are

\frac{1}{x}

\frac{1}{x\ln 10}

✅

\frac{\ln 10}{x}

\frac{10}{x}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The general formula $\frac{d}{dx}\log_{b}x = \frac{1}{x\ln b}$ applies. For base 10, $\ln 10$ is a constant, so the derivative becomes $\frac{1}{x\ln 10}$ . This differs from $\frac{1}{x}$ for the natural log, making option B correct.

Q4. Apply the chain rule to find $\frac{d}{dx}\bigl[\ln(\sqrt{x^{2}+1})\bigr]$ .

\frac{x}{\sqrt{x^{2}+1}}

\frac{1}{2\sqrt{x^{2}+1}}

\frac{2x}{x^{2}+1}

\frac{x}{x^{2}+1}

✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Let $u=\sqrt{x^{2}+1}$ . Then $\ln u$ ' = $\frac{1}{u}u'$ . Since $u = (x^{2}+1)^{1/2}$ , $u' = \frac{x}{\sqrt{x^{2}+1}}$ . Multiplying gives $\frac{x}{x^{2}+1}$ . This matches option D, demonstrating careful chain‑rule application.

Q5. Using the limit definition, which limit evaluates to the derivative of $\ln x$ at a point $x>0$ ?

\displaystyle\lim_{h\to0}\frac{\ln(x+h)-\ln x}{h}

✅

\displaystyle\lim_{h\to0}\frac{e^{x+h}-e^{x}}{h}

\displaystyle\lim_{h\to0}\frac{x^{h}-1}{h}

\displaystyle\lim_{h\to0}\frac{\ln h}{h}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The derivative of a function $f$ at $x$ is defined as $\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ . Substituting $f(x)=\ln x$ yields the expression in option A, which directly computes $\frac{1}{x}$ . The other limits correspond to different functions.

Q6. What is $\frac{d}{dx}\bigl[\ln(5x^{2}+3)\bigr]$ ?

\frac{5x}{5x^{2}+3}

\frac{10x}{5x^{2}+3}

✅

\frac{2x}{5x^{2}+3}

\frac{10x}{x^{2}+3}

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Apply the chain rule: derivative of $\ln u$ is $u'/u$ . Here $u=5x^{2}+3$ and $u'=10x$ . Hence $\frac{d}{dx}\ln(5x^{2}+3)=\frac{10x}{5x^{2}+3}$ . This corresponds to option B.

Q7. Why is the natural logarithm base $e$ preferred when differentiating logarithmic functions?

A.It yields integer results

B.It simplifies integration only

C.It makes logarithms periodic

D.Its derivative formula is

\frac{1}{x}

✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: When the base is $e$ , the derivative of $\log_{e}x = \ln x$ reduces to the simple form $\frac{1}{x}$ . This eliminates the extra constant $\ln b$ that appears for other bases, streamlining calculations in calculus, which justifies option D.

Q8. If $g'(x)=\frac{2}{x\ln 3}$ , which function could represent $g(x)$ (up to an additive constant)?

2\log_{3}x

✅

\frac{2}{\ln x}

\ln(3x^{2})

\frac{2}{x^{2}\ln 3}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Integrating $g'(x)=\frac{2}{x\ln 3}$ gives $g(x)=\frac{2}{\ln 3}\int\frac{1}{x}dx =\frac{2}{\ln 3}\ln x +C =2\log_{3}x +C$ . Therefore the antiderivative matches option A, showing the connection between derivative and logarithm base conversion.

Q9. Find the derivative of $h(x)=\frac{\ln x}{\ln (x^{2})}$ .

A.0 ✅

\frac{1}{x\ln x}

\frac{1}{2x\ln x}

\frac{2}{x}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Since $\ln (x^{2}) = 2\ln x$ , the function simplifies to $h(x)=\frac{\ln x}{2\ln x}= \frac{1}{2}$ for $x>0, x\neq1$ . The derivative of a constant is zero, so the correct answer is option A. Recognizing the simplification is essential for this higher‑order problem.

Q10. For which base $b$ does the derivative of $\log_{b}x$ equal $\frac{1}{x}$ ?

A.b=10

B.b=2

C.b=e ✅

D.Any

b>0

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The general derivative formula is $\frac{d}{dx}\log_{b}x = \frac{1}{x\ln b}$ . To equal $\frac{1}{x}$ , the denominator $\ln b$ must be 1, which occurs only when $b=e$ . Thus option C correctly identifies the natural base.

Q11. Differentiate $f(x)=\ln (x)^{2}$ (i.e., $(\ln x)^{2}$ ). Which expression is correct?

\frac{2}{x\ln x}

\frac{2\ln x}{x}

✅

\frac{\ln x}{2x}

\frac{2}{x}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Treat $f(x) = (\ln x)^{2}$ as a composition. Using the chain rule, $f'(x)=2\ln x \cdot \frac{1}{x}= \frac{2\ln x}{x}$ . This matches option B, demonstrating proper application of differentiation rules for composite functions.

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