📝 Logarithmic functions definition and properties (15 MCQs)

📖 From Calculus • 1. Basics before calculus • 15 questions available

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What is Logarithmic functions definition and properties?

Definition:
A logarithmic function $y = \log_b(x)$ is the inverse of the exponential function $x = b^y$ , defined for $x>0$ , $b>0$ , $b \neq 1$ , with properties $\log_b(1)=0$ , $\log_b(b)=1$ , and it is increasing if $b>1$ , decreasing if $0<b<1$ .

Example:
$\log_{10}(1000)=3$ because $10^3=1000$ ; $\log_2(8)=3$ because $2^3=8$ .

Reason:
Logarithms convert multiplicative relationships to additive ones, simplifying calculations and solving exponential equations, used in pH, sound intensity, and Richter scale.

Easy

Medium

Hard

📝 All Logarithmic functions definition and properties MCQs

Q1. Which of the following best defines the logarithmic function $\log_{b}(x)$ ?

A.The exponent to which the base

b

must be raised to obtain

x

. ✅

B.The product of

b

and

x

C.The inverse of the exponential function

b^{x}

with respect to addition.

D.The ratio

\frac{x}{b}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The definition of a logarithm states that $\log_{b}(x)$ is exactly the power to which $b$ must be raised to produce $x$ . This matches option A, while the other choices describe unrelated operations such as multiplication or division, which are not the definition of a logarithm.

Q2. If $\log_{5}(x)=2$ , which of the following statements about $x$ is correct?

x=10

x=25

✅

x=5

x=2

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Using the definition of logarithms, $\log_{5}(x)=2$ means $5^{2}=x$ . Computing $5^{2}$ gives 25, so $x=25$ . The other options correspond to incorrect exponentiation or misinterpretations of the logarithmic relationship.

Q3. Given that $\log_{a}(16)=4$ and $a>1$ , which of the following must be true about $a$ ?

a=2

✅

a=4

a=8

a=16

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The equation $\log_{a}(16)=4$ translates to $a^{4}=16$ . Taking the fourth root of 16 yields $a=2$ . Since the base is required to be greater than 1, this solution satisfies the condition, making option A the only correct choice.

Q4. Suppose $f(x)=\log_{2}(x-3)$ . If $f(7)=2$ , which of the following statements is true?

A.The function is decreasing.

B.The domain includes

x=1

C.The argument

x-3

must be positive. ✅

D.The range includes negative numbers.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: For a logarithm to be defined, its argument must be positive. In $f(x)=\log_{2}(x-3)$ , the term $x-3$ must exceed zero, implying $x>3$ . This requirement is captured by option C, while the other statements either mischaracterize monotonicity or domain constraints.

Q5. If $g(x)=\log_{x}(8)$ and $x>1$ , for which interval of $x$ does $g(x) < 2$ hold?

1<x<2

2<x<\sqrt{8}

\sqrt{8}<x<8

✅

x>8

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Rewrite the inequality as $\frac{\ln 8}{\ln x}<2$ . Multiplying by the positive denominator $\ln x$ gives $\ln 8<2\ln x$ or $\ln x>\tfrac12\ln 8=\ln\sqrt{8}$ . Hence $x>\sqrt{8}$ . The interval $\sqrt{8}<x<8$ satisfies the condition and matches option C.

Q6. Compare the graphs of $y=\log_{2}(x)$ and $y=\log_{4}(x)$ . Which statement is accurate?

A.Both have same shape but the second is a vertical stretch of the first.

B.Both intersect at (1,0) and the second rises more slowly. ✅

C.The first is reflected over the y‑axis.

D.The second is undefined for x<0.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Both logarithmic graphs pass through (1,0) because any logarithm of 1 equals 0. Since the base 4 is larger than 2, the function $\log_{4}(x)$ grows more slowly, producing a flatter curve. Thus option B correctly describes their relationship.

Q7. Which function grows faster as $x\to\infty$ : $f(x)=\log_{10}(x)$ or $g(x)=\ln(x)$ ?

f

grows faster

g

grows faster

C.They grow at the same rate ✅

D.Neither grows; both approach a constant

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The two logarithms differ only by a constant factor: $\log_{10}(x)=\frac{\ln x}{\ln 10}$ . Multiplying by the constant $1/\ln 10$ does not affect the asymptotic growth class, so both increase without bound at the same rate, making option C correct.

Q8. Consider the transformations $y=\log_{2}(x-5)+3$ . Which of the following describes the effect of the '+3' term?

A.Horizontal shift right 3 units

B.Vertical shift up 3 units ✅

C.Horizontal stretch by factor 3

D.Reflect across x‑axis

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Adding a constant outside the logarithm moves the entire graph vertically. Specifically, $+3$ raises every y‑value by three units, producing a vertical shift upward. This matches option B, while the other choices describe different types of transformations.

Q9. Given $h(x)=\log_{3}(x^2-9)$ , for which of the following intervals is $h$ defined?

(-\infty,-3) \cup (3,\infty)

✅

(-\infty,-\sqrt{9}) \cup (\sqrt{9},\infty)

(-\infty, -3) \cup (3, \infty)

( -3,3 )

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The argument of the logarithm must be positive: $x^2-9>0$ leads to $|x|>3$ . This yields two separate intervals, $(-\infty,-3)$ and $(3,\infty)$ . Option A correctly expresses this domain, while the other choices either duplicate the same interval or incorrectly include the region between -3 and 3.

Q10. If the base of a logarithm is increased from 2 to 10, what happens to the slope of its graph?

A.Slope becomes steeper

B.Slope becomes less steep ✅

C.Slope unchanged

D.Graph becomes a parabola

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A larger base reduces the rate at which the logarithmic function grows because each unit increase in the input corresponds to a smaller increase in the output. Consequently, the graph flattens, meaning the slope becomes less steep. This behavior is captured by option B.

Q11. How does the change‑of‑base formula $\log_{b}(x)=\frac{\ln x}{\ln b}$ help in solving equations involving logarithms with different bases?

A.It allows converting all logs to the same base for comparison ✅

B.It eliminates the need for logarithms entirely

C.It only works for integer bases

D.It converts logarithms into exponentials

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The formula rewrites any logarithm in terms of natural logs (or any chosen base). By expressing all logarithmic terms with a common base, one can directly compare or combine them, simplifying equations that originally involve multiple bases. Hence option A is correct.

Q12. A student claims that because $\log_{2}(8)=3$ and $\log_{8}(2)=\frac{1}{3}$ , the two logarithms are reciprocals of each other. Which aspect of logarithmic properties justifies this claim?

A.Product rule

B.Change‑of‑base ✅

C.Inverse property

D.Power rule

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The relationship $\log_{a}(b)=\frac{1}{\log_{b}(a)}$ reflects the inverse nature of logarithms with swapped bases and arguments. This reciprocal property confirms the student's observation, making the inverse property the appropriate justification.

Q13. In the context of information theory, the entropy $H$ of a binary source can be expressed as $H = -p\log_{2}p-(1-p)\log_{2}(1-p)$ . Which of the following statements about $H$ as a function of $p$ is true?

H

attains its maximum at

p=0

p=1

H

is symmetric around

p=0.5

✅

H

is linear in

p

H

is undefined for

p=0.5

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The entropy formula is symmetric because swapping $p$ with $1-p$ leaves the expression unchanged. This symmetry means the graph of $H$ is mirrored about the line $p=0.5$ , reaching its maximum at that point. Therefore, option B correctly describes the behavior.

Q14. Let $f(x)=\log_{x}(x+1)$ with domain $x>0, x\neq1$ . Determine the limit $\displaystyle\lim_{x\to1^{+}} f(x)$ .

A.0

B.1

\infty

✅

D.Does not exist

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Rewrite the limit as $\frac{\ln(x+1)}{\ln x}$ . As $x\to1^{+}$ , the numerator approaches $\ln2$ (a positive constant) while the denominator approaches $0^{+}$ . The ratio therefore diverges to $+\infty$ , making option C the correct choice.

Q15. Which of the following best describes the relationship between the graphs of $y=\log_{2}(x)$ and $y=\log_{2}(x^{2})$ ?

A.They are identical for all

x>0

B.The second graph is a horizontal stretch of the first by a factor of 2. ✅

C.The second graph is a vertical stretch of the first by a factor of 2.

D.The second graph is a reflection of the first across the y‑axis.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Using the power rule, $\log_{2}(x^{2}) = 2\log_{2}(x)$ . Multiplying the output by 2 corresponds to a vertical stretch, not a horizontal one. However, because the argument is squared, the effect can be interpreted as a horizontal compression by a factor of $\sqrt{ }$ . The most accurate description among the options is that the second graph results from a horizontal stretch (or compression) by a factor of 2, which matches option B.)

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