Pure Mathematics - Exponentials and Logarithms Practice Questions
A-Levels · A-Level Mathematics · 150 free MCQs with instant results and detailed explanations.
150
Total
78
Easy
57
Medium
15
Hard
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Sample Questions from Pure Mathematics - Exponentials and Logarithms
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Q1
Easy
If y = 2^x, what is the value of x when y = 16?
A.
4
B.
2
C.
8
D.
3
Show Answer & Explanation
Correct Answer: A
y = 2^x means that when y = 16, we can express 16 as 2^4, hence x = 4.
Q2
Easy
Which of the following is the correct property of logarithms?
A.
log_a(b * c) = log_a(b) + log_a(c)
B.
log_a(b + c) = log_a(b) + log_a(c)
C.
log_a(b - c) = log_a(b) - log_a(c)
D.
log_a(b/c) = log_a(b) + log_a(c)
Show Answer & Explanation
Correct Answer: A
The property states that the logarithm of a product is equal to the sum of the logarithms of the factors.
Q3
Easy
If e^x = 5, what is x approximately equal to?
A.
ln(5)
B.
5
C.
2.3
D.
e^5
Show Answer & Explanation
Correct Answer: A
To solve for x, we take the natural logarithm of both sides: x = ln(5). This is the definition of the natural logarithm.
Q4
Medium
If f(x) = e^(2x), what is f'(x)?
A.
2e^(2x)
B.
e^(2x)
C.
e^(x)
D.
2xe^(2x)
Show Answer & Explanation
Correct Answer: A
The derivative of e^(kx) is k e^(kx). Here k=2, thus f'(x) = 2e^(2x).
Q5
Medium
The equation 3^x = 81 can be solved using logarithms. What is the value of x?
A.
4
B.
3
C.
2
D.
5
Show Answer & Explanation
Correct Answer: A
Since 81 = 3^4, we can equate x to 4, thus x = 4.
Q6
Medium
If log_a(16) = 4, what is the value of a?
A.
2
B.
4
C.
8
D.
16
Show Answer & Explanation
Correct Answer: A
Using the definition of logarithms, we have a^4 = 16. Since 16 = 2^4, it follows that a = 2.
Q7
Medium
If f(x) = log(3x + 2), find f'(x).
A.
(3 / (3x + 2))
B.
(1 / (3x + 2))
C.
(3 / (x + 2))
D.
(1 / (3x))
Show Answer & Explanation
Correct Answer: A
Using the chain rule, the derivative f'(x) = (1/(3x+2)) * 3 = 3/(3x+2).
Q8
Hard
Consider the function f(x) = e^(2x) + 3e^x. What is the minimum value of f(x) for all x ∈ ℝ?
A.
3
B.
6
C.
9
D.
12
Show Answer & Explanation
Correct Answer: B
To find the minimum value of f(x), we can rewrite it as f(x) = e^x (e^x + 3). The minimum occurs when e^x is minimized, which is at x = 0, yielding f(0) = 3. However, the function goes to infinity as x increases or decreases. Therefore, we check critical points by differentiating: f'(x) = 2e^(2x) + 3e^x, setting that to zero gives minimum at x = 0, resulting in f(0) = 6.
Q9
Hard
If log_b(x) + log_b(y) = 5 and log_b(xy) = 8, what is the value of b?
A.
32
B.
64
C.
16
D.
4
Show Answer & Explanation
Correct Answer: A
Using log properties, log_b(xy) = log_b(x) + log_b(y). Hence, log_b(xy) = 5 = 8, which gives two equations: 1) log_b(x) + log_b(y) = 5; 2) log_b(x) + log_b(y) = 8. Therefore, we equate these to find log_b(x) = 8 - log_b(y). Solving gives b = 32.
Q10
Hard
If the function f(x) = e^(2x) + 3e^x + 2 is transformed into g(x) by factoring, what is g(x)?
A.
g(x) = (e^x + 1)(e^x + 2)
B.
g(x) = (e^x + 2)(e^x + 1)
C.
g(x) = (e^(x) + 3)(e^(x) + 1)
D.
g(x) = (e^(x) + 1)(e^(x) + 3)
Show Answer & Explanation
Correct Answer: A
Factoring f(x) = e^(2x) + 3e^x + 2 yields g(x) = (e^x + 1)(e^x + 2) as both (e^x + 1) and (e^x + 2) multiply correctly back to the original function.
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Pure Mathematics - Exponentials and Logarithms — A-Levels A-Level Mathematics Practice Questions Online
This page contains 150 practice MCQs for the chapter Pure Mathematics - Exponentials and Logarithms in A-Levels A-Level Mathematics. The questions are organized by difficulty — 78 easy, 57 medium, 15 hard — so you can choose the right level for your preparation.
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