Logarithms

Test Name	No. of Questions	Marks (of each)	Time	Take Test
Test-1	12	3	40 Minutes	Take Test
Test-2	12	3	40 Minutes	Take Test
Test-3	13	3	40 Minutes	Take 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 $\frac{a}{m} = \frac{b}{n} \neq \frac{c}{p}$ : 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 $\frac{a}{m} \neq \frac{b}{n}$ : They have different slopes and hence must intersect at some point, resulting in a unique solution.

If $\frac{a}{m} = \frac{b}{n} = \frac{c}{p}$ : 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.

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

Question 2.

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

Question 3.

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

log4 (3/2)

log4 (7/2)

log4 (7)

log4(23/2)

Question 4.

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

Question 5.

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

Question 6.

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

Question 7.

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

4 < a < 5

2 < a < 3

a > 5

3 < a < 4

Question 8.

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

log₂(1/5)

–log₂ (1/3)

–log₂ (1/5)

log₂ (1/3)

Question 9.

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

Question 10.

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

-1

-0.5

Question 11.

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

Question 12.

If $\log_{a}30$ = A, $\log_{a}5/3$ = -B and $\log_{2}a$ = 1/3, then $\log_{3}a$ equals

(A + B)/2 - 3

2/(A + B) - 3

2/(A + B - 3)

(A + B - 3)/2

Question 13.

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

250

150

100

Question 14.

The real root of the equation $2^{6x}$ + $2^{3x+2}$ - 21 = 0 is

\frac{\log_{2}(3)}{3}

{\log_{2}(9)}

\frac{{\log_{2}(7)}}{3}

{\log_{2}(27)}

Question 15.

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

A < 1/16

A < 1/8

A > 1/8

A > 1/16

Question 16.

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

-3 ≤ x ≤ 3

1 ≤ x ≤ 2

1 ≤ x ≤ 3

-1 ≤ x ≤ 3

Question 17.

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

\log_{5}(5/3)

\log_{5}9

\log_{5}8

1 +

\log_{5}(3/5)

Question 18.

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

\log_{2}8

\log_{6}16

\log_{6}8

\log_{4}16

Question 19.

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

Question 20.

If p^₃ = q⁴ = r⁵ = s⁶, then the value of log_s(pqr) is equal to

47/10

16/5

24/5

Question 21.

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

Question 22.

$\frac{1}{\log_{2}100}$ - $\frac{1}{\log_{4}100}$ + $\frac{1}{\log_{5}100}$ - $\frac{1}{\log_{10}100}$ + $\frac{1}{\log_{20}100}$ - $\frac{1}{\log_{25}100}$ + $\frac{1}{\log_{50}100}$ ?

-4

1/2

Question 23.

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/2

Question 24.

The value of log_0.008√5 + log_√381 – 7 is equal to:

1/3

2/3

5/6

7/6

Question 25.

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

0 < x < 3

23 < x < 30

x > 30

3 < x < 23

Question 26.

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

Question 27.

If log_yx = a × log_zy = b × log_xz = ab, then which of the following pairs of values for (a, b) is not possible?

-2 , 1/2

1,1

0.4, 2.5

2, 2

Question 28.

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

$log_{x}$ (x/y) + $log_{y}$ (y/x) can never be

-1

-0.5

Question 29.

Let u = (log₂x)² – 6 log₂ x + 12 where x is a real number. Then the equation x^u = 256, has

no solution for x

exactly one solution for x

exactly two distinct solutions for x

exactly three distinct solutions for x

Question 30.

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

Question 31.

If 1/3 $\log_{3} M + 3$ \log_{3} N = 1 + $\log_{0.008}$ 5 , then

M^{9}

= 9/N

N^{9}

= 9/M

M^{3}

= 3/N

N^{9}

= 3/M

Question 32.

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

1/100

1/1000

None of these

Question 33.

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)) ?

log[

\frac{}{3}

]

Question 34.

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 $\log_{2}x$ = $\sqrt{x}$
II. x ≤ 10 km

Question 35.

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

None of these

Question 36.

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

None of these

Question 37.

log₆ 216 $\sqrt{6}$ is

3/2

7/2

None of these

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Tips To Solve CAT Logarithms

CAT Logarithms PDF

Question 1.

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

Question 2.

For some positive ral number x, if log⁡3(x)\log_{\sqrt{3}}(x)log3​​(x) + log⁡x25log⁡x(0.008)\frac{\log_{x}25}{\log_{x}(0.008)}logx​(0.008)logx​25​ = 16/ 3 , then the value of log⁡3(3x2)\log_{3}(3x^{2})log3​(3x2) is

Question 3.

For a real number x, if 1/ 2 , log⁡3(2x−9)log⁡34\frac{\log_{3}(2^{x}-9)}{\log_{3}4}log3​4log3​(2x−9)​ , and log⁡5(2x+17/2)log⁡54\frac{\log_{5}(2^{x}+17/2)}{\log_{5}4}log5​4log5​(2x+17/2)​ are in arithmetic progression, then the common difference is

Question 4.

The number of distinct integer values of n satisfying 4−log⁡2n3−log⁡4n\frac{4-\log_{2}n}{3-\log_{4}n}3−log4​n4−log2​n​ < 0, is

Question 5.

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

Question 6.

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

Question 7.

For a real number a, if log⁡15a+log⁡32alog⁡15a+log⁡32a\frac{\log_{15}a+\log_{32}a}{\log_{15}a+\log_{32}a}log15​a+log32​alog15​a+log32​a​ = 4, then a must lie in the range

Question 8.

If y is a negative number such that 2y2log⁡35=5log⁡232^{y^{2}\log_{3}5}=5^{\log_{2}3}2y2log3​5=5log2​3 , then y equals

Question 9.

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

Question 10.

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

Question 11.

2X4X8X16(log⁡24)2(log⁡48)3(log⁡816)4\frac{2X4X8X16}{(\log_{2}4)^{2}(\log_{4}8)^{3}(\log_{8}16)^{4}}(log2​4)2(log4​8)3(log8​16)42X4X8X16​ equals .

Question 12.

If log⁡a30\log_{a}30loga​30 = A, log⁡a5/3\log_{a}5/3loga​5/3 = -B and log⁡2a\log_{2}alog2​a = 1/3, thenlog⁡3a\log_{3}alog3​a equals

Question 13.

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

Question 14.

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

Question 15.

Let A be a real number. Then the roots of the equation x2x^{2}x2 - 4x - log⁡2A\log_{2}Alog2​A = 0 are real and distinct if and only if

Question 16.

If x is a real number, then log⁡e(4x−x2)/3\sqrt{\log_{e}(4x-x^2)/3}loge​(4x−x2)/3​ is a real number if and only if

Question 17.

If x is a positive quantity such that 2x = 3log⁡523^{\log_{5}2}3log5​2 , then x is equal to

Question 18.

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

Question 19.

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

Question 20.

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

Question 21.

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

Question 22.

Question 23.

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:

Question 24.

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

Question 25.

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

Question 26.

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

Question 27.

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

Question 28.

If x ≥ y and y > 1, then the value of the expression logxlog_{x}logx​ (x/y) + logylog_{y}logy​ (y/x) can never be

Question 29.

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

Question 30.

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

Question 31.

If 1/3 log⁡3M+3\log_{3} M + 3 log3​M+3\log_{3} N = 1 + log⁡0.008\log_{0.008}log0.008​ 5 , then

Question 32.

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

Question 33.

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)) ?

Question 34.

Question 35.

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

Question 36.

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

Question 37.

log6 216 6\sqrt{6}6​ is

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

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

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

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

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

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

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

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

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

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

If $\log_{a}30$ = A, $\log_{a}5/3$ = -B and $\log_{2}a$ = 1/3, then $\log_{3}a$ equals

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

The real root of the equation $2^{6x}$ + $2^{3x+2}$ - 21 = 0 is

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

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

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

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

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

If p^₃ = q⁴ = r⁵ = s⁶, then the value of log_s(pqr) is equal to

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

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:

The value of log_0.008√5 + log_√381 – 7 is equal to:

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

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

If log_yx = a × log_zy = b × log_xz = ab, then which of the following pairs of values for (a, b) is not possible?

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

$log_{x}$ (x/y) + $log_{y}$ (y/x) can never be

Let u = (log₂x)² – 6 log₂ x + 12 where x is a real number. Then the equation x^u = 256, has

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

If 1/3 $\log_{3} M + 3$ \log_{3} N = 1 + $\log_{0.008}$ 5 , then

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

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)) ?

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

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

log₆ 216 $\sqrt{6}$ is