 # CAT Questions | CAT Algebra Questions: Logarithm

### CAT Verbal Ability

Logarithm or logs is one of the easiest topics to cover in the Quantitative Aptitude section for any MBA entrance exam. Log CAT questions assess a candidate on 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 CAT questions based on logarithm and also in other MBA exams such as NMAT, XAT, SNAP, etc

## CAT Quantitative Aptitude questions | CAT Algebra: Logarithm

1. If log4 5 = (log4 y) (log6√5), then y equals ______________                          [CAT 2020]

log45 = log4y × log6√5

log45 ÷ log4y = logy5 = log6√5 = k (Let’s assume this as k)

6k = √5 —> On squaring this {6k}2 = 5

log45 = log4y × log6√5

logy5 = log6√5

logy5 = k

yk = 5

{6k}2 = yk = 5

{62}k = yk

36k = yk

y = 36

2. log9 (3log2 (1 + log3 (1 + 2log2x))) = 1212. Find x.                                       [CAT 2020]

A. 4     B. 1212     C. 1       D. 2

log9 (3log2 (1 + log3 (1 + 2log2x)) = 1212

3log2(1 + log3(1 + 2log2x)) = 91/2 = 3

log2(1 + log3(1 + 2log2x) = 1

1 + log3(1 + 2log2x) = 2

log3(1 + 2log2x) = 1

1 + 2log2x = 3

2log2x = 2

log2x = 1

x = 2

3. If 22x+4 – 17 × 2x+1 = –4, then which of the following is true?                                          [CAT 2019]

A. x is a positive value

B. x is a negative value

C. x can be either a positive value or a negative value

D. None of these

2x+4 – 17 * 2x+1 = – 4

=> 2x+1 = y

22x+2 = y2

22(22x+2) – 17 * 2x+1 = –4

4y2 – 17y + 4 = 0

4y2 – 16y – y + 4 = 0

4y (y – 4) – 1 (y – 4) = 0

y = 1414 or 4

2x+1 = 1414 or 4

=> x + 1 = 2 or – 2

x = 1 or – 3

4. The real root of the equation 26x + 23x+2 − 21 = 0 is                                 [CAT 2019]

A. log(base2)3 / 2     B. log(base2)9     C. log(base2)27     D. log(base2)7 / 3

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

## Basic concepts

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 on Logarithm 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

• Characteristic of a number greater than unity for a common base is positive and is 1 less than the number of digits in the integral part.For example : Characteristic of log 1000 = 3 which is 1 less than the number of digits in 1000.

• 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.

• There are 2 types of logarithms that are commonly used on the basis of bases:

Natural logarithm : base of the number is “e” .

Common logarithm : Base of the number is 10 . When the base is not mentioned , it can be taken as 10.

• log( x – y ) ≠ logx – logy

• log( x + y ) ≠ logx + logy

## Important Logarithm Identities:

• 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:

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

## How to deal with Logarithm 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 from MBAP Live Lecture Recording (Basic) on basic concept and MBAP lecture Assignment. You can also practice Log questions from MBAP CAT E Book (Concept Theory) study material.

• 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 log and also learn initial few values of logarithm table

• Practice beginner-level log questions from MBAP CAT E Book (Practice Questions).

Level – 2

• Move on to more complex problems on logarithms, attempt beginner and intermediate level mock on concepts of Logarithm.

• Solve previous year CAT log questions for practice from MBAP Topic wise Previous Year CAT Question or MBAP Previous Year CAT paper and time yourself. Do not get stuck on one question and try to solve easy questions first.

• Keep attempting mocks to check your performance and try to get acquainted with different types of logarithms problems.

• Topic-wise mocks, provided by MBAP, can be utilized to enhance your performance.

Level – 3

• For advanced level preparation, start practicing questions on logarithm from the MBAP CAT Advance E books or 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. Counselling Session
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