Logarithms — CAT Previous-Year Questions

34 previous-year questions on Logarithms from CAT, with full solutions. Practise free — check answers as you go; sign in to save your progress.

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34 questions

Logarithms · CAT PYQs

CAT 2024 Slot 1 · QA

Q1.

If x is a positive number such that 4 log₁₀x + 4 log₁₀₀x + 8 log₁₀₀₀x = 13, then the greatest integer not exceeding x, is

CAT 2024 Slot 2 · QA

Q2.

If a, b and c are postive real numbers such that a > 10 ≥ b ≥ c and $\frac{\log_{8} (a + b)}{\log_{2} c}$ + $\frac{\log_{27} (a - b)}{\log_{3} c}$ = $\frac{2}{3}$ , then the greatest possible integer value of a is

CAT 2023 Slot 1 · QA

Q3.

If x and y are positive real numbers such that log_x (x² + 12) = 4 and 3log_y x = 1, then x + y equals?

CAT 2023 Slot 2 · QA

Q4.

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

CAT 2023 Slot 3 · QA

Q5.

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

CAT 2022 Slot 2 · QA

Q6.

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

CAT 2021 Slot 1 · QA

Q7.

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

CAT 2021 Slot 2 · QA

Q8.

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

CAT 2021 Slot 3 · QA

Q9.

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

CAT 2020 Slot 1 · QA

Q10.

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

CAT 2020 Slot 1 · QA

Q11.

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

CAT 2020 Slot 2 · QA

Q12.

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

CAT 2020 Slot 3 · QA

Q13.

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

CAT 2020 Slot 3 · QA

Q14.

If log_a30 = A, log_a(5/3) = -B and log₂a = 1/3, then log₃a equals

CAT 2019 Slot 1 · QA

Q15.

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

CAT 2019 Slot 2 · QA

Q16.

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

CAT 2018 Slot 1 · QA

Q17.

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

CAT 2018 Slot 1 · QA

Q18.

If log₁₂81 = p, then $3 (\frac{4 - p}{4 + p})$ is equal to

CAT 2018 Slot 1 · QA

Q19.

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

CAT 2018 Slot 2 · QA

Q20.

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

CAT 2018 Slot 2 · QA

Q21.

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

CAT 2018 Slot 2 · QA

Q22.

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

CAT 2017 Slot 1 · QA

Q23.

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

CAT 2017 Slot 2 · QA

Q24.

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

CAT 2006 · QA

Passage / Data

Answer the following question based on the information given below.

An airline has a certain free luggage allowance and charges for excess luggage at a fixed rate per kg. Two passengers, Raja and Praja have 60 kg of luggage between them, and are charged Rs. 1200 and Rs. 2400 respectively for excess luggage. Had the entire luggage belonged to one of them, the excess luggage charge would have been Rs. 5400.

Q25.

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

CAT 2005 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Q26.

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

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

CAT 2004 · QA

Passage / Data

Answer the following question based on the information given below.

In an examination, there are 100 questions divided into three groups A, B and C such that each group contains at least one question. Each question in group A carries 1 mark, each question in group B carries 2 marks and each question in group C carries 3 marks. It is known that the questions in group A together carry at least 60% of the total marks.

Q27.

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

CAT 2003 Slot 2 · QA

Passage / Data

Answer the following question based on the information given below.

Consider three circular parks of equal size with centres at A₁, A₂ and A₃ respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles A₁A₂A₃, B₁B₂B₃ and C₁C₂C₃, as shown. Three sprinters A, B, and C begin running from points A₁, B₁ and C₁ respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.

Q28.

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

CAT 2003 Slot 2 · QA

Passage / Data

Answer the following question based on the information given below.

A string of three English letters is formed as per the following rules:

The first letter is any vowel.
The second letter is m, n or p.
If the second letter is m, then the third letter is any vowel which is different from the first letter.
If the second letter is n, then the third letter is e or u.
1. If the second letter is p, then the third letter is the same as the first letter.

Q29.

If log₁₀ x - log₁₀ $\sqrt{x}$ = 2 log_x 10, then a possible value of x is given by:

CAT 2003 Slot 2 · QA

Passage / Data

Answer the following question based on the information given below.

A string of three English letters is formed as per the following rules:

The first letter is any vowel.
The second letter is m, n or p.
If the second letter is m, then the third letter is any vowel which is different from the first letter.
If the second letter is n, then the third letter is e or u.
1. If the second letter is p, then the third letter is the same as the first letter.

Q30.

What is the sum of n terms in the series

$\log m + \log \frac{m^{2}}{n} + \log \frac{m^{3}}{n^{2}} + \log \frac{m^{4}}{n^{3}} + \dots + \log \frac{m^{n}}{n^{n - 1}} ?$

CAT 1999 · QA

Passage / Data

Directions: Answer the questions based on the following information.
Recently, Ghosh Babu spent his winter vacation on Kyakya Island. During the vacation, he visited the local casino where he came across a new card game. Two players, using a normal deck of 52 playing cards, play this game. One player is called the ‘dealer’ and the other is called the ‘player’. First, the player picks a card at random from the deck. This is called the base card. The amount in rupees equal to the face value of the base card is called the base amount. The face values of ace, king, queen and jack are ten. For other cards the face value is the number on the card. Once the ‘player’ picks a card from the deck, the ‘dealer’ pays him the base amount. Then the ‘dealer’ picks a card from the deck and this card is called the top card. If the top card is of the same suit as the base card, the ‘player’ pays twice the base amount to the ‘dealer’. If the top card is of the same colour as the base card (but not the same suit), then the ‘player’ pays the base amount to the ‘dealer’. If the top card happens to be of a different colour than the base card, the ‘dealer’ pays the base amount to the ‘player’.
Ghosh Babu played the game four times. First time he picked eight of clubs and the ‘dealer’ picked queen of clubs. Second time, he picked ten of hearts and the ‘dealer’ picked two of spades. Next time, Ghosh Babu picked six of diamonds and the ‘dealer’ picked ace of hearts. Lastly, he picked eight of spades and the ‘dealer’ picked jack of spades. Answer the following questions based on these four games.

Q31.

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

CAT 1997 · QA

Passage / Data

Answer the next 3 questions based on the following information.

There are 60 students in a class. These students are divided into three groups A, B and C of 15, 20 and 25 students each. The groups A and C are combined to form group D.

Q32.

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

CAT 1994 · QA

Passage / Data

Answer the next 3 questions based on the information given below:

Ghoshbabu is staying at Ghosh Housing Society, Aghosh Colony, Dighospur, Calcutta. In Ghosh Housing Society 6 persons read daily Ganashakti and 4 read Anand Bazar Patrika; in his colony there is no person who reads both. Total number of persons who read these two newspapers in Aghosh Colony and Dighospur is 52 and 200 respectively. Number of persons who read Ganashakti in Aghosh Colony and Dighospur is 33 and 121 respectively; while the persons who read Anand Bazar Patrika in Aghosh Colony and Dighospur are 32 and 117 respectively.

Q33.

If log₇ log₅ $(\sqrt{x + 5} + \sqrt{x})$ = 0, find the value of x.

CAT 1994 · QA

Passage / Data

Answer the next 3 questions based on the information given below:

Alphonso, on his death bed, keeps half his property for his wife and divide the rest equally among his three sons Ben, Carl and Dave. Some years later Ben dies leaving half his property to his widow and half to his brothers Carl and Dave together, shared equally. When Carl makes his will he keeps half his property for his widow and the rest he bequeaths to his younger brother Dave. When Dave dies some years later, he keeps half his property for his widow and the remaining for his mother. The mother now has Rs. 1,575,000.

Q34.

log₆ 216 $\sqrt{6}$ is