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
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.
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:
- The integral part is called Characteristic
- 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:
- 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.
- 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.
- log( x – y ) ≠ logx – logy
- 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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Words From Our Students
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.
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