logo
iconCall Us
Test NameNo. of QuestionsMarks (of each)TimeTake Test
Test-112340 MinutesTake Test
Test-212340 MinutesTake Test
Test-313340 MinutesTake Test

Tips To Solve CAT Logarithms

- Fundamentals of this concept are useful in solving the questions of the other topics by assuming the unknown values as variables. Make sure to cover other inter-related concepts of CAT syllabus. All the inter-related concepts need to be covered to have a good foundation in concepts.

- Be careful of silly mistakes in this topic, as that is how students generally lose marks here. The number of equations needed to solve the given problem equals the number of variables. A linear equation is an equation which gives a straight line when plotted on a graph.

- If you are confused, enrolling in CAT online coaching will help you a long way.

- Linear equations can be of one variable or two variables, or three variables.

  • Let a, b, c and d be constants, and x, y, and z are variables. A general form of a single variable linear equation is ax + b = 0.
  • A general form of two-variable linear equation is ax + by = c.
  • A general form of three-variable linear equation is ax + by + cz = d.

CAT Logarithms PDF

To help CAT aspirants in their preparation, we have made a comprehensive formula PDF containing all the important linear equations that are essential. This PDF includes all the necessary formulas, techniques, and examples required to solve linear equations efficiently. Click on the link below to download the Linear equations formula PDF.

1. Linear Equations Formulae: Solving Linear Equations

For equations of the form ax + by = c and mx + ny = p, find the LCM of b and n.

Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1.

2. Linear Equations Formulae: Straight Lines

Equations with 2 variables: Consider two equations ax + by = c and mx + ny = p. Each of these equations represents two lines on the x-y coordinate plane. The solution of these equations is the point of intersection.

If am=bncp: This means that both the equations have the same slope but different intercepts, and hence are parallel to each other. There is no point of intersection and no solution.

If ambn: They have different slopes and hence must intersect at some point, resulting in a unique solution.

If am=bn=cp: The two lines have the same slope and intercept. Hence, they are the same lines. As they have infinite points common between them, there are infinitely many solutions possible.

Question 1.

Coachify Logo

If x and y are positive real numbers such that logx(x2+12)\log_{x}(x^2+12) = 4 and 3logyx=13\log_{y}x=1, then x + y equals?

A
11
B
68
C
20
D
10

Question 2.

Coachify Logo

For some positive ral number x, if log3(x)\log_{\sqrt{3}}(x) + logx25logx(0.008)\frac{\log_{x}25}{\log_{x}(0.008)} = 16/ 3 , then the value of log3(3x2)\log_{3}(3x^{2}) is

A
B
C
D

Question 3.

Coachify Logo

For a real number x, if 1/ 2 , log3(2x9)log34\frac{\log_{3}(2^{x}-9)}{\log_{3}4} , and log5(2x+17/2)log54\frac{\log_{5}(2^{x}+17/2)}{\log_{5}4} are in arithmetic progression, then the common difference is

A
log4 (3/2)
B
log4 (7/2)
C
log4 (7)
D
log4(23/2)

Question 4.

Coachify Logo

The number of distinct integer values of n satisfying 4log2n3log4n\frac{4-\log_{2}n}{3-\log_{4}n} < 0, is

A
B
C
D

Question 5.

Coachify Logo

If 5 -log10(1+x)\log_{10}(\sqrt{1+x}) + 4log10(1x)\log_{10}(\sqrt{1-x}) = log10(11x2)\log_{10}(\frac{1}{\sqrt{1-x^{2}}}) , then 100x equals

A
B
C
D

Question 6.

Coachify Logo

log2 [3 + log3[4+log4(x1)]\log_{3}[{4 + log_{4}(x - 1)}]] - 2 = 0, then 4x equals

A
B
C
D

Question 7.

Coachify Logo

For a real number a, if log15a+log32alog15a+log32a\frac{\log_{15}a+\log_{32}a}{\log_{15}a+\log_{32}a} = 4, then a must lie in the range

A
4 < a < 5
B
2 < a < 3
C
a > 5
D
3 < a < 4

Question 8.

Coachify Logo

If y is a negative number such that 2y2log35=5log232^{y^{2}\log_{3}5}=5^{\log_{2}3} , then y equals

A
log₂(1/5)
B
–log₂ (1/3)
C
–log₂ (1/5)
D
log₂ (1/3)

Question 9.

Coachify Logo

If log₄ 5 = (log₄ y)(log₆ √5), then y equals

A
B
C
D

Question 10.

Coachify Logo

The value of⁡ loga(ab)+logb(ba)\log_{a}(\frac{a}{b}) + \log_{b}(\frac{b}{a}) , for 1 < a ≤ b cannot be equal to

A
0
B
-1
C
1
D
-0.5

Question 11.

Coachify Logo

2X4X8X16(log24)2(log48)3(log816)4\frac{2X4X8X16}{(\log_{2}4)^{2}(\log_{4}8)^{3}(\log_{8}16)^{4}} equals .

A
B
C
D

Question 12.

Coachify Logo

If loga30\log_{a}30 = A, loga5/3\log_{a}5/3 = -B and log2a\log_{2}a = 1/3, thenlog3a\log_{3}a equals

A
(A + B)/2 - 3
B
2/(A + B) - 3
C
2/(A + B - 3)
D
(A + B - 3)/2

Question 13.

Coachify Logo

Let x and y be positive real numbers such that log5(x + y) + log5(x - y) = 3, and log2y - log2x = 1 - log23. Then xy equals

A
250
B
150
C
100
D
25

Question 14.

Coachify Logo

The real root of the equation 26x2^{6x} + 23x+22^{3x+2}- 21 = 0 is

A
log2(3)3\frac{\log_{2}(3)}{3}
B
log2(9){\log_{2}(9)}
C
log2(7)3\frac{{\log_{2}(7)}}{3}
D
log2(27){\log_{2}(27)}

Question 15.

Coachify Logo

Let A be a real number. Then the roots of the equation x2x^{2} - 4x - log2A\log_{2}A = 0 are real and distinct if and only if

A
A < 1/16
B
A < 1/8
C
A > 1/8
D
A > 1/16

Question 16.

Coachify Logo

If x is a real number, then loge(4xx2)/3\sqrt{\log_{e}(4x-x^2)/3} is a real number if and only if

A
-3 ≤ x ≤ 3
B
1 ≤ x ≤ 2
C
1 ≤ x ≤ 3
D
-1 ≤ x ≤ 3

Question 17.

Coachify Logo

If x is a positive quantity such that 2x = 3log523^{\log_{5}2} , then x is equal to

A
1+log5(5/3)\log_{5}(5/3)
B
log59\log_{5}9
C
log58\log_{5}8
D
1 +log5(3/5)\log_{5}(3/5)

Question 18.

Coachify Logo

If log1281 = p, then 3((4-p)/(4+p)) is equal to

A
log28\log_{2}8
B
log616\log_{6}16
C
log68\log_{6}8
D
log416\log_{4}16

Question 19.

Coachify Logo

If log2(5 + log3 a) = 3 and log5(4a + 12 + log2 b) = 3, then a + b is equal to

A
40
B
67
C
59
D
32

Question 20.

Coachify Logo

If p3 = q4 = r5 = s6, then the value of logs(pqr) is equal to

A
47/10
B
16/5
C
24/5
D
1

Question 21.

Coachify Logo

The smallest integer n for which 4n > 1719 holds, is closest to

A
33
B
37
C
39
D
35

Question 22.

Coachify Logo

1log2100\frac{1}{\log_{2}100} - 1log4100\frac{1}{\log_{4}100} + 1log5100\frac{1}{\log_{5}100} - 1log10100\frac{1}{\log_{10}100} + 1log20100\frac{1}{\log_{20}100}  - 1log25100\frac{1}{\log_{25}100} + 1log50100\frac{1}{\log_{50}100} ?

A
-4
B
10
C
0
D
1/2

Question 23.

Coachify Logo

Suppose, log3x = log12y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log6G is equal to:

A
√a
B
2a
C
a/2
D
a

Question 24.

Coachify Logo

The value of log0.008√5 + log√381 – 7 is equal to:

A
1/3
B
2/3
C
5/6
D
7/6

Question 25.

Coachify Logo

If x is a real number such that log35 = log5(2 + x), then which of the following is true?

A
0 < x < 3
B
23 < x < 30
C
x > 30
D
3 < x < 23

Question 26.

Coachify Logo

If log (2a × 3b × 5c) is the arithmetic mean of log (22 × 33 × 5), log (26 × 3 × 57), and log (2 × 32 × 54), then a equals

A
B
C
D

Question 27.

Coachify Logo

If logyx = a × logzy = b × logxz = ab, then which of the following pairs of values for (a, b) is not possible?

A
-2 , 1/2
B
1,1
C
0.4, 2.5
D
2, 2

Question 28.

Coachify Logo

If x ≥ y and y > 1, then the value of the expression

logxlog_{x} (x/y) + logylog_{y} (y/x) can never be

A
-1
B
-0.5
C
0
D
1

Question 29.

Coachify Logo

Let u = (logx)2 – 6 log2 x + 12 where x is a real number. Then the equation xu = 256, has

A
no solution for x
B
exactly one solution for x
C
exactly two distinct solutions for x
D
exactly three distinct solutions for x

Question 30.

Coachify Logo

If log3 2, log3 (2x − 5), log3 (2x − 7/2) are in arithmetic progression, then the value of x is equal to

A
5
B
4
C
2
D
3

Question 31.

Coachify Logo

If 1/3 log3M+3\log_{3} M + 3 \log_{3} N = 1 + log0.008\log_{0.008} 5 , then

A
M9M^{9} = 9/N
B
N9N^{9} = 9/M
C
M3M^{3} = 3/N
D
N9N^{9} = 3/M

Question 32.

Coachify Logo

If log10 x - log10 x^(1/2) = 2 logx 10, then a possible value of x is given by:

A
10
B
1/100
C
1/1000
D
None of these

Question 33.

Coachify Logo

What is the sum of n terms in the series

log m + log (m^2 / n) + log (m^3 / n^2) + log (m^4 / n^3) .......+ log (m^n / n^(n-1)) ?

A
log[3\frac{}{3}]
B
C
D

Question 34.

Coachify Logo

Directions: Each question is followed by two statements I and II. Mark:
1. if the question can be answered by any one of the statements alone, but cannot be answered by using the other statement alone.
2. if the question can be answered by using either statement alone.
3. if the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
4. if the question cannot be answered even by using both the statements together.

What is the distance x between two cities A and B in integral number of kilometres?
I. x satisfies the equation log2x\log_{2}x = x\sqrt{x}
II. x ≤ 10 km

A
1
B
2
C
3
D
4

Question 35.

Coachify Logo

If log2 [log7 (x2 - x + 37)] = 1, then what could be the value of ‘x’?

A
3
B
5
C
4
D
None of these

Question 36.

Coachify Logo

If log7 log5 ((x+5)^(1/2) + x^(1/2)) = 0, find the value of x.

A
1
B
0
C
2
D
None of these

Question 37.

Coachify Logo

log6 216 6\sqrt{6} is

A
3
B
3/2
C
7/2
D
None of these

Explore Our Learning Zones

Access a vast collection of previous year questions to sharpen your exam preparation, all for free!

Whatsapp
Logo
CAT Batch Discount?
YoutubeInstagramTelegramWhatsappPhone