300+ Most Important Algebra Questions for CAT
Logarithms
Logarithms are a regularly tested and interconnected topic in Algebra for CAT 2026. This page includes targeted CAT algebra important questions built around repeated log identities and equation-based CAT problems. The focus is on accuracy, concept clarity, and speed. For stronger performance, also revise Functions, Quadratic Equations, and Inequalities, as CAT often mixes these concepts with logarithms.
x > 8
4 < x < 5
6 < x < 7
7 < x < 8
Q3. What is the value of x, if logx (log108 (√3 + √75)) = –1/2?
Q5. How many integer values of x satisfy the equation |2 + log1/7 x| = 1 + |1 + log1/7 x|?
5
6
7
4
Q6. Let loga (logb (logc p)) = 0, where a, b and c assume distinct values among, 4, 8 and 16 only. If the product of all possible values of ‘p’ is equal to 2n, then what is the value of ‘n’?
Q7. Let xyz = 1081 and log10 x × log10 y + log10 y × log10 z + log10 z × log10 x = 468, where x, y and z are positive numbers, then (log10 x)2 + (log10 y)2 + (log10 z)2 is divisible by:
54
33
53 x 33
22 x 34
Q8. If log3(4 + log2P) = 2 and log2(log3Q – 4) = 1 then the value of P2 – Q is:
305
295
285
270
Q9. Find the value of x, if log4 32 + log4 128 = log2 16 + log2 √x
Q11. The numerical values of the surface area and the volume of a cuboid are equal. If the edge lengths are log2 x, log3 x, log5 x, then the value of ‘x’ is:
Q12. If x = log10 (1 + 2 + 3 + … + n) + log10 2, where n is a natural number. Find the number of possible values of n for which 2 < x < 3.
Q13. For how many values of ‘k’ are log a, log ka and log (4a) are in an arithmetic progression?
Q14. Find the number of solutions of the equation log3 (x + 5) = 6 – x.
0
1
2
More than 2
-3/2
3/2
-2/3
2/3
Q16. If log (x4y) = 1; log (xy3) = 2, what is the value of log(xy)?
8/11
7/13
8/33
3/11
0
1
2
3
Q18. If log2 x + log2 y ≥ 6, then what is the least value of x + y?
1 + log5 3 – log5 2
1 + log5 3 – 2log5 2
1 + log3 5 – 4log5 2
1 + log5 3 – 4log5 2
Q21. It is given that X and Y are two distinct natural numbers. Which of the following can be the value of ‘a’ given that loga X > loga Y implies X < Y?
10
5
1
0.5
3 log 40
5 log 42
3 log 42
5 log 40
1
2
8
4
8
9
7
3
10 5/18
10 2/9
10 9/2
10 18/5
3/2
1/4
1/2
2
10 -1/5
10 1/5
5 -1/2
10 -3/5
17
28
34
1
11
10
8
16
4
9
16
Either (1) or (3)
1
3
2
0
Q36. If loga 36 = 1.44, and log2 a = 3.2, then find the value of log2 32a + loga 144.
9.64
10.265
10.015
9.9525
1
2
3
More than 3
Q38. Given that logx (logy (logz p)) = 0, where each of x, y and z can assume distinct values among 7, 49 and 2401 only. If the product of all possible values of ‘p’ is represented in the form of 7n, then what is the value of ‘n’?
8400
7490
7140
9450
1
2
3
4
27 - 2
2(27 - 1)
2(27 + 1)
27 + 1
12
192
24
512
n2
2n
4n
2n2
logx logy
4(logx – logy)
2(logx + logy)
None of the above
1
5
5/4
25/12
1
2
4
Cannot be determined
-1/2
5/8
3/8
7/2
