Logarithm

Logarithms or logs is one of the easiest topics to cover for the Quantitative Aptitude section in any MBA entrance exam. It assesses a candidate with his/her ability to calculate the log of a given number n as an exponent to which another fixed number, the base b, must be raised, to produce that number n.

A candidate can easily expect 2-3 questions based on logarithm in CAT and other MBA exams such as NMAT, XAT, SNAP, etc.

Exams which use contain this topic:

CAT – Log questions in CAT may range from very easy to very high level of difficulty. You may expect 2-3 questions from this topic. CAT has been consistently assessing candidates on Logarithms and exponents

YearNo of QuestionsLevel of Difficulty

 

2019

 

Slot 11Easy
Slot 22Easy to moderate

 

2018

 

Slot 13Moderate
Slot 23Moderate to High

NMAT – The Log questions in NMAT usually range from easy to moderate and are a must attempt. One must not miss out on these questions.

 

YearNo of QuestionsLevel of Difficulty
20192Easy to Moderate
20182Moderate

 

List of basic concepts of Logs:

Logarithm questions are generally direct, but the level of difficulty may be increased by adding the concept of the number of digits.

Listed below are a few concepts that may help you gain insight into the type of questions asked:

If ax  = N , then, x = log of N to the base a and x = logaN . In other words, it represents the power to which a number must be raised.

Suppose we are asked the result if ‘x’ is multiplied by itself ‘y’ times; then your answer would be x = x*x*x*…. y (times). This can also be written as x^y. This is also known as ‘x raised to the power of y’

The log of a number comprises 2 parts:

  1. The integral part is called Characteristic
  2. The decimal part is called Mantissa

For example, Log 27 = 3 Log 3 = 3*0.4771 = 1.4313

 

In this case, the characteristic is 1 and the mantissa is 0.4313

Important formulae to remember:

  • logxx = 1
  • logx1 = 0
  • logxab = b logx a
  • logx(mn) = logx m + logx n
  • logxax = a
  • logxm = (logy n) x (loga n)
  • logx(mn) = logx m + logx n
  • logx(m/n) = logx m – logx n

Key points to note:

  1. The characteristic of a number greater than unity for a common base is positive and is 1 less than the number of digits in an integral part. For example, the Characteristic of log 1000 = 3 which is 1 less than the number of digits in 1000.
  2. For a number between 0 and 1, the characteristic is negative, and its magnitude is 1 more than the number of zeros after the decimal point. For example: Characteristic of log 0.001 = -3.
  3. log( x – y ) ≠ logx – logy
  4. log( x + y ) ≠ logx + logy

Some Questions from Previous Papers

CAT 2019 Slot 1:  Let x and y be positive real numbers such that log5 (x + y) + log5 (x y) = 3, and log2 y log2 x = 1 log2 3. Then xy equals 

1. 150 

2. 100 

3. 25 

4. 250

 

Ans. Given, log(base5) (x + y) + log(base5) (x − y) = 3

Or log(base5) (x + y)*(x-y) =3

Or x^2 –y^2 = 5^3 = 125————-1)

log(base2) y − log(base2) x = 1 − log(base2) 3= log(base2) 2 – log(base2) 3

log(base2) y/x = log(base2) 2/3

y/x = 2/3

y = 2x/3

from eq 1) x^2 – (2x/3)^2 = 125

x^2 – (4x^2/9) = 125

5x^2 = 125*9 or x^2 = 225

x = 15

y= 2x/3 = 30/3 = 10

 

xy = 15*10 =150

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CAT 2019 Slot 2:  If x is a real number, then is a real number if and only if 

1. 1≤x≤2 

2. -3≤x≤3 

3. -1≤x≤3 

4. 1≤x≤3

Ans. As we know that any value under square root must be greater than 0. So Log(base e) 4x-x^2/3 ≥ 0 

So, 4x-x^2/3≥ 1 

x^2-4x +3 ≤ 0 

 

On solving, we get S belongs to [1,3]

 

CAT 2019 Slot 2: The real root of the equation 2 6x + 2 3x+2 – 21 = 0 is 

1. log(base2)3 / 2 

2. log(base2)9 

3. log(base2)27 

4. log(base2)7 / 3

Ans. Let 2^(3x) = k 

So given equation 2^6x + 2^(3x+2) – 21 =0 

Or (2^3x)^2 + 4*2^3x -21 =0 

Or k^2 + 4k -21 =0 

(k+7)*(k-3) =0 

k = -4 or 3 

k= -4 is not possible 

so k =3 

or 2^3x = 3 

taking log of both sides 3x * log 2 = log 3 

3x = log 3 / log 2 

3x = log(base2)⁡3 

Or x = (log(base2⁡)3) /3 

Option a) (log(base2)⁡3) /3

 

XAT: Find the value of log10 10 + log10 10 2 + ….. + log10 10 n

1. n^2 + 1

2. n^2 − 1

3. (n^2 +n)/2 n(n+1)/3

4. (n^2 +n)/2

log10 10 + log10 10 2 + ….. + log10 10 n

Since loga = 1

log10 10 + log10 10 2 + ….. + log10 10 n=1+2+…n

n(n+1) 2 (n +n)

D is the correct answer.

 

How to deal with that topic preparation

Preparing for Logarithm will require a basic understanding of the key concepts and formulae along with patience and a knack for learning. Here are some Level-wise preparatory guidelines to follow:

Level – 1

  • Learn the basic concepts of logarithm thoroughly.
  • Learn Speed calculation: For an effective and quick calculation, be thorough with tables till 20, memorize squares till 30, and cubes till 15.
  • Memorize the properties of the log.
  • Practice beginner-level questions.

Level – 2

  • Move on to more complex problems, attempt beginner and intermediate level mock on concepts of Logarithm.
  • Solve previous year CAT questions on Logs and time yourself. Do not get stuck on one question and try to solve easy questions first.
  • Keep attempting mocks to check your performance.
  • Topic-wise mocks, provided by MBAP, can be utilized to enhance your performance.

Level – 3

  • For advanced level preparation, start practicing questions from the book – How to Prepare for Quantitative Aptitude for the CAT, authored by Arun Sharma.
  • Questions in Arun Sharma are categorized into Level of Difficulty (LOD), based upon your preparation level, start attempting 3 or 4 questions daily.