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📝 Exponential and logarithmic functions continuity (14 MCQs)

📖 From Calculus • 2. Limits and Continuity an Introduction • 14 questions available

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What is Exponential and logarithmic functions continuity?

Definition:
Exponential functions $f(x) = a^x$ with $a > 0, a \neq 1$ are continuous everywhere on $\mathbb{R}$ , and their inverses, logarithmic functions $\log_a x$ , are continuous on $(0, \infty)$ . This continuity follows from their definitions as continuous extensions of rational powers and the inverse function theorem, ensuring smooth behavior without breaks over their domains.

Example:
Evaluate $\lim_{x \to 2} e^{x-1}$ and $\lim_{x \to 4} \ln x$ .
Solution: $e^{x-1}$ continuous, so $\lim = e^{1}$ ; $\ln x$ continuous at $4$ , so $\lim = \ln 4$ .

Reason:
Continuity of these functions is essential for modeling growth and decay processes in biology, finance, and physics, and for solving equations involving exponentials and logarithms without abrupt changes.

Easy

Medium

Hard

📝 All Exponential and logarithmic functions continuity MCQs

Q1. According to theorem 1.6.3, on which interval is the function $b^{x}$ continuous for a base $b>0$ with $b\\neq1$ ?

A.\

(- \\infty , \\infty )\

✅

B.\

(0,\\infty )\

C.\

(- \\infty ,0)\

D.\

[0,\\infty )\

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The theorem explicitly states that the exponential function $b^{x}$ has no restrictions on the exponent, so it is continuous on the entire real line \ $(- \\infty , \\infty )\$ .

Q2. If $f(x)=\\ln x\$ and $g(x)=e^{x}\$ , which of the following statements is always true?

A.Both

f

and

g

are continuous for all real

x\

f

is continuous on \

(0,\\infty )\

while

g

is continuous on \

(-\\infty ,\\infty )\

. ✅

C.Both functions are discontinuous at

x=0\

D.Only

g

is continuous on \

(0,\\infty )\

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The natural logarithm requires a positive argument, giving continuity on \ $(0,\\infty )\$ , whereas the exponential function is defined for every real number, so the combined statement B is correct.

Q3. Compare the domains of continuity for $b^{x}$ and \ $\\log_{b}x\$ . Which statement correctly describes the difference?

A.Both are continuous on \

(0,\\infty )\

B.Both are continuous on \

(-\\infty ,\\infty )\

C.\

b^{x}\

is continuous on \

(-\\infty ,\\infty )\

while \

\\log_{b}x\

is continuous only on \

(0,\\infty )\

. ✅

D.\

b^{x}\

is continuous only on \

(0,\\infty )\

while \

\\log_{b}x\

is continuous on \

(-\\infty ,\\infty )\

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The exponential function accepts any real exponent, giving continuity on the whole line. The logarithm needs a positive argument, restricting its continuity to \ $(0,\\infty )\$ .

Q4. Let \ $f(x)=\\begin{cases}\\dfrac{e^{x}-1}{x}, & x\\neq0\\\\ 1, & x=0\\end{cases}\$ . Which of the following best explains why \ $f\$ is continuous at \ $x=0\$ ?

A.The numerator and denominator are both continuous, so their quotient is continuous.

B.The limit \

\\displaystyle\\lim_{x\\to0}\\frac{e^{x}-1}{x}=1\

matches the defined value at \

x=0\

. ✅

C.Since \

e^{x}\

is continuous, any expression involving it is also continuous.

D.The function is a polynomial, hence continuous everywhere.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Evaluating the limit gives \ $1\$ ; because the piecewise definition assigns the same value at \ $x=0\$ , the function has no jump, ensuring continuity at that point.

Q5. Consider \ $h(x)=\\ln(x)+b^{x}\$ defined for all real \ $x\$ . On which interval is \ $h\$ continuous?

A.\

(-\\infty ,0)\

B.\

(0,\\infty )\

✅

C.\

(- \\infty ,\\infty )\

D.\

[0,\\infty )\

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The logarithm requires a positive argument, so the sum is defined only for \ $x>0\$ . The exponential part poses no restriction, thus the overall continuity is on \ $(0,\\infty )\$ .

Q6. For a fixed base \ $b>0,\\ b\\neq1\$ , evaluate the continuity of \ $p(x)=\\frac{b^{x}-1}{x}\$ at \ $x=0\$ . Which statement is correct?

A.The function has a removable discontinuity at \

x=0\

because the limit does not exist.

B.The limit \

\\displaystyle\\lim_{x\\to0}\\frac{b^{x}-1}{x}=\\ln b\

exists, so defining \

p(0)=\\ln b\

makes the function continuous. ✅

C.The function is continuous at \

x=0\

without any modification.

D.The function is discontinuous at every point because the denominator vanishes.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Using the derivative definition of \ $b^{x}\$ at zero gives the limit \ $\\ln b\$ . Assigning this value removes the hole, so the extended function becomes continuous at \ $0\$ .

Q7. Which principle best justifies that \ $\\log_{b}x\$ is continuous on \ $(0,\\infty )\$ ?

A.Composition of continuous functions: \

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

. ✅

B.Since \

b^{x}\

is continuous, its inverse must also be continuous everywhere.

C.The definition of logarithm as an integral guarantees continuity.

D.Continuity follows from the Intermediate Value Theorem alone.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Expressing \ $\\log_{b}x\$ as a constant multiple of \ $\\ln x\$ shows it inherits the continuity of \ $\\ln x\$ on its domain, because dividing by the non‑zero constant \ $\\ln b\$ does not affect continuity.

Q8. Determine the continuity of \ $f(x)=\\ln(x^{2}+1)\$ over the real line.

A.Continuous only for \

x>0\

B.Continuous only for \

x\\neq0\

C.Continuous on \

(-\\infty ,\\infty )\

✅

D.Discontinuous at \

x=0\

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The inner polynomial \ $x^{2}+1\$ is always positive, so the logarithm is defined for every real \ $x\$ . Both the polynomial and the natural logarithm are continuous, making the composition continuous everywhere.

Q9. Which of the following functions is NOT continuous at \ $x=1\$ ?

A.\

b^{x-1}\

B.\

\\log_{b}(x-1)\

✅

C.\

\\frac{b^{x}-b}{x-1}\

D.\

\\ln(x)+b^{x}\

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: \ $\\log_{b}(x-1)\$ requires its argument to be positive, i.e., \ $x>1\$ . At \ $x=1\$ the argument is zero, where the logarithm is undefined, causing a discontinuity.

Q10. Let \ $g(x)=\\ln x-\\frac{x}{e}\$ on \ $(0,\\infty )\$ . Using the Intermediate Value Theorem, which conclusion is guaranteed?

A.There exists a root \

c\

in \

(0,\\infty )\

where \

g(c)=0\

B.\

g(x)\

attains a maximum at \

x=1\

C.\

g(x)\

is always positive. ✅

D.No root exists because \

g\

is decreasing.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Evaluating \ $g(1)=-1/e<0\$ and \ $g(e)=0\$ shows a sign change on a continuous interval, so the IVT ensures a point \ $c\\in(1,e)\$ with \ $g(c)=0\$ .

Q11. Given that \ $b^{x}\$ is continuous on \ $\\mathbb{R}\$ , what can be concluded about its inverse \ $\\log_{b}x\$ ?

A.It is discontinuous everywhere.

B.It is continuous on \

(-\\infty ,\\infty )\

C.It is continuous on \

(0,\\infty )\

. ✅

D.It is continuous only at integer points.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The inverse of a continuous, strictly monotonic function is continuous on the image of the original function. Since \ $b^{x}\$ maps \ $\\mathbb{R}\$ onto \ $(0,\\infty )\$ , its inverse is continuous exactly on that interval.

Q12. For the function \ $f(x)=\\tan^{-1}x+\\ln x\$ with denominator \ $x^{2}-4\$ forming \ $F(x)=\\dfrac{f(x)}{x^{2}-4}\$ , on which intervals is \ $F\$ continuous?

A.\

(0,2)\

and \

(2,\\infty )\

✅

B.\

(-\\infty ,0)\

and \

(0,2)\

C.\

(-\\infty ,2)\

only

D.\

(2,\\infty )\

only

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Both \ $\\tan^{-1}x\$ and \ $\\ln x\$ are continuous for \ $x>0\$ . The denominator vanishes at \ $x=\\pm2\$ ; only the positive zero affects the domain, yielding continuity on \ $(0,2)\$ and \ $(2,\\infty )\$ .

Q13. Simplify the composite function \ $k(x)=\\log_{b}\\bigl(b^{x^{2}}\\bigr)\$ and state its continuity.

A.It simplifies to \

x^{2}\

and is continuous on \

(-\\infty ,\\infty )\

. ✅

B.It simplifies to \

b^{x^{2}}\

and is continuous only on \

(0,\\infty )\

C.It simplifies to \

\\ln(x^{2})\

and is discontinuous at \

x=0\

D.It simplifies to \

x^{2}\

but is continuous only for \

x>0\

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Using the identity \ $\\log_{b}(b^{y})=y\$ gives \ $k(x)=x^{2}\$ . Polynomials are continuous everywhere, so the composite function is continuous on the entire real line.

Q14. Suppose \ $f(x)=\\ln(x)\$ and \ $g(x)=b^{x}\$ with \ $b>1\$ . If a sequence \ $\\{x_n\\}\$ converges to \ $c>0\$ , which statement about the limits of \ $f(x_n)\$ and \ $g(x_n)\$ is correct?

A.Both limits exist and equal \

f(c)\

and \

g(c)\

respectively.

B.Only \

\\displaystyle\\lim_{n\\to\\infty}g(x_n)\

exists.

C.Only \

\\displaystyle\\lim_{n\\to\\infty}f(x_n)\

exists. ✅

D.Neither limit is guaranteed to exist.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Continuity of \ $\\ln x\$ on \ $(0,\\infty )\$ and of \ $b^{x}\$ on \ $\\mathbb{R}\$ implies that the limit of each function at a convergent sequence equals the function value at the limit point, giving both limits as \ $f(c)\$ and \ $g(c)\$ .

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