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Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

Document Description: Important Formulas: Logarithms for CAT 2022 is part of Quantitative Aptitude (Quant) preparation. The notes and questions for Important Formulas: Logarithms have been prepared according to the CAT exam syllabus. Information about Important Formulas: Logarithms covers topics like Logarithm, Properties of Logarithms and Important Formulas: Logarithms Example, for CAT 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Important Formulas: Logarithms.

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Table of contents
Logarithm
Properties of Logarithms
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Logarithm

  • If a is a positive real number, other than 1 and am = x, then we write: m = loga x and we say that the value of log x to the base a is m.
  • Examples: (i) 103 1000 ⇒ log10 1000 = 3.
    (ii) 34 = 81 ⇒ log3 81 = 4.
    Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
    (iv) (. 1)2 = 01 ⇒ log(.1) .01 = 2. 


Properties of Logarithms

1. log a (xy) = loga x + loga y

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

3. logx x = 1 

4. loga 1 = 0 

5. loga (xn) = n(loga x)

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT


Common Logarithms

  • Logarithms to the base 10 are known as common logarithms.
  • The logarithm of a number contains two parts, namely 'characteristic' and 'mantissa'.

1. Characteristic

The internal part of the logarithm of a number is called its characteristic.
Case I: When the number is greater than 1.

  • In this case, the characteristic is one less than the number of digits in the left of the decimal point in the given number.

Case II: When the number is less than 1.

  • In this case, the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative.
  • Instead of -1, -2 etc. we write Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT (two bar), etc.
    Examples:
    Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

2. Mantissa:

The decimal part of the logarithm of a number is known is its mantissa. For mantissa, we look through log table.


Question 1: If log2X + log4X = log0.25√6 and x > 0, then x is:

A. 6-1/6

B. 61/6

C. 3-1/3

D. 61/3

Correct Answer is Option (A).

  • log2x + log4x = log0.25√6
    We can rewrite the equation as:  
    Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
     log2x * 3 = 2log0.25√6
     log2x3 = -log46 
    Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
    Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
     2log2x3 = -log26
  • 2log2x3 + log26 = 0
    log26X6 = 0
  • 6x6 = 1 
    x6 = 1/6
    Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
  • The question is "If log2X + log4X = log0.25 √6 and x > 0, then x is"
  • Hence, the answer is "6-1/6".

Question 2: log9 (3log2 (1 + log3 (1 + 2log2x))) = 1/2. Find x. 

A. 4

B. 1/2

C. 1

D. 2

Correct Answer is Option (D).

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

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 

The question is "Find x."

Hence, the answer is "2".


Question 3: If 22x+4 – 17 × 2x+1 = –4, then which of the following is true? 

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 

Correct Answer is Option (C).

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 + y = 0 
4y (y – 4) – 1 (y – 4) = 0 

y = 1/4 or 4

2x+1 = 1/4 or 4

⇒ x + 1 = 2 or – 2
x = 1 or – 3

The question is "which of the following is true?"

Hence, the answer is "x can be either a positive value or a negative value".


Question 4: If log1227 = a, log916 = b, find log8108 

A. Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

B. Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

C. Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

D. Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

Correct Answer is Option (D).

log8108 = log8(4 * 27) 
log8108 = log84 + log827 
 log84 = 2/3

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
log827 = 2 * log16
log916 = b 
log169 = 1/b

log827 = 2/b

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

The question is "find log8108."

Hence, the answer is Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT


Question 5: Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT If a, b are integers such that x = a, and x = b satisfy this inequation, find the maximum possible value of a – b. 

A. 214

B. 216

C. 200

D. 203

Correct Answer is Option (A).

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

log3x = y

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

y ∈ (3, 5)
3 < log3x < 5
27 < x < 243
Therefore max ( a – b) will be when a = 242 and b = 28. Therefore, max(a – b) = 214.

The question is "find the maximum possible value of a – b."

Hence, the answer is "214".


Question 6: log5x = a (This should be read as log X to the base 5 equals a) log20x = b. What is logx10?

A. Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
B. (a + b) * 2ab

C. Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
D. Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

Correct Answer is Option (A).

Given, log5x = a
log20x = b
logx5 = 1/a

logx20 = 1/b
Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT
Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

The question is "What is logx10?"

Hence, the answer is Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT


Question 7: log3x + logx3 = 17/4. Find x.

A. 34

B. 31/8

C. 31/4

D. 31/3

Correct Answer is Option (C).

log3x + logx3 = 17/4

Let y = log3x
We know that logx3 = Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

Hence logx3 = 1/y

Thus the equation can be written asImportant Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

4y2 + 4 = 17y
4y2 + 4 - 17y = 0
Solving the above equation we get y = 4 or 1/4

If y = 4
log3x = 4
Then x = 34
If y = 1/4

log3x = 1/4

Then x = 31/4
The question is "Find x."

Hence, the answer is "34".

Question 8: logxy + logyx2 = 3. Find logxy3. 

A. 4

B. 3

C. 31/2

D. 31/16

Correct Answer is Option (B).

logxy + logyx2 = 3
Let a = logxy 

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

logyx2 = 2logyx
We know that logyx =Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

Hence form above logyx = 1/a

Now rewritting the equation logxy + logyx2 = 3 

Using a we get Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

i.e., a2 - 3a + 2 = 0
Solving we get a = 2 or 1 
If a = 2, Then logxy = 2 and logyx3 = 3
logxy = 3 * 2 = 6
Or
If a = 1, Then logxy = 1 and logyx3 = 3
logxy = 3 * 1 = 3

The question is "Find logxy3."

Hence, the answer is "3".


Question 9: log2 4 * log4 8 * log8 16 * ……………nth term = 49, what is the value of n? 

A. 49

B. 48

C. 34

D. 24

Correct Answer is Option (B).

First, the nth term of L.H.S need to be defined by observing the pattern :-
It is log(2n) 2.2n 
Given,
log2 4 * log4 8 * log8 16 * ……………log(2n) 2.2n = 49
Whenever solving a logarithm equation, generally one should approach towards making the base same.
Making the base 2:- 

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

log(2n) 2.2n = 49
log(2n) 2 + log(2n) 2n = 49
1 + n = 49
n = 48

The question is "what is the value of n?"

Hence, the answer is "48".


Question 10: If 33 + 6 + 9 + ……… 3x =Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT what is the value of x? 

A. 3

B. 6

C. 7

D. 11

Correct Answer is Option (D).

First of all, let us define the xth term. 

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

Whenever you encounter a distinctive number such as one given in R.H.S of above equation, always try to find its significance in the context of question. 

In this case L.H.S has 3a, so Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT must be some form of 3a. 

With little hit and trial, you may find Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

33(1 + 2 + 3 + ...X) = 3 -3 * -66
 33 * 3x(x+1)/2 = 33*66 

 Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

x(x+1) = 132
Solving this equation for x > 0, we get x = 11.
You should directly be able to see that 132 = 11 * 12 => x= 11
And avoid wasting time solving the complete equation.

The question is "what is the value of x?"

Hence, the answer is "11".

Question 11: x, y, z are 3 integers in a geometric sequence such that y - x is a perfect cube. Given, log36x2 + log6√y + log216y1/2z = 6. Find the value of x + y + z. 

A. 189

B. 190

C. 199

D. 201

Correct Answer is Option (A).

Let us begin with simplifying the equation:-
 log62x2 + log6y1/2 + 3log63y1/2z = 6 

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

log6x + log6y1/2y1/2z = 6
log6xyz = 6
xyz = 66 

Given x,y,z is in G.P. Let x = a, y = ab, z = ab2
⇒ xyz = a3b3 = (ab)3 
(ab)3 = (62)3
Possible values of (a,b) satisfying the equation :-
(1, 36), (2, 18), (3, 12), (4, 9), (9, 4), (12, 3), (18, 2), (36, 1)
Given y-x is a perfect cube
⇒ ab-a is perfect cube
⇒ a(b-1) is perfect cube
Only possible when (a, b) = (9, 4)
∴ x = 9 , y = 36 , z = 144
∴ x + y + z = 9 + 36 + 144 = 189

The question is "Find the value of x + y + z."

Hence, the answer is "189".

Question 1210log(3 - 10logy) = log2(9 - 2y), Solve for y. 

A. 0

B. 3

C. 0 and 3

D. none of these

Correct Answer is Option (D).

Before beginning to simplify the equation, don’t forget that anything inside a log cannot be negative 
10log(3-y) = log2(9 - 2y) (y > 0)…………………………………(1)
3 - y = log2(9 - 2y) (Therefore, 3 - y > 0 =) (y < 3)) ……………………… (2)
23-y = 9 - 2y
2y = t 

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

⇒ 8 = 9t –t2
⇒ t2 - 9t + 8 = 0
⇒ t2 - t - 8t - 8 = 0
⇒ t(t - 1) - 8(t - 1) = 0
⇒ t = 1, 8
Therefore, 2y = 1 and 2y = 8
⇒ y = 0 and y = 3
However, from inequalities (1) and (2), y cannot take either of these value.

The question is "Solve for y."

Hence, the answer is "none of these".

Question 13: 46+12+18+24+…+6x = (0.0625)-84, what is the value of x? 

A. 7

B. 6

C. 9

D. 12

Correct Answer is Option (A).

Take right side expression, 

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

= (4-2)-84
= 4168
Take left side expression 
46+12+18+24+…+6x = 46(1+2+3+4+x)
= 46 * 4(1+2+3+4+…+x)
= 46 * 4x(x+1)/2 (using the formula for sum of natural numbers from 1 to x)
Equating left and right side expresssions, we get 46 * 4x(x+1)/2 = 4168
Or 46 * 4x(x+1)/2 = 46*28

Important Formulas: Logarithms Notes | Study Quantitative Aptitude (Quant) - CAT

or x (x + 1) = 56
Solving for x we get, x = 7

The question is "what is the value of x?"

Hence, the answer is "7".
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