Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

Quantitative Aptitude (Quant)

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The document Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev is a part of the UPSC Course Quantitative Aptitude (Quant).
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1. 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.

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

(iv). (. 1)2 = 01 ⇒ log(.1) .01 = 2. 

2. Properties of Logarithms: 

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

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

3. logx x = 1 

4. loga 1 = 0 

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

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

Common Logarithms:

Logarithms to the base 10 are known as common logarithms.

The logarithm of a number contains two parts, namely 'characteristic' and 'mantissa'.

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 Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev (two bar), etc. 

Examples:-

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

Mantissa:

The decimal part of the logarithm of a number is known is its mantissa. For mantissa, welook 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

Answer. A

Explanation.

log2x + log4x = log0.25√6
We can rewrite the equation as:  

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev
 log2x * 3 = 2log0.25√6
 log2x3 = -log46 
Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev
Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

 2log2x3 = -log26
2log2x3 + log26 = 0
log26X6 = 0
6x6 = 1 

x6 = 1/6
Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

The question is "If log2X + log4X = log0.25 √6 and x > 0, then x is"

Hence, the answer is "6-1/6".

Choice A is the correct answer. 

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

A. 4

B. 1/2

C. 1

D. 2

Answer. D

Explanation.

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".

Choice D is the correct answer.

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 

Answer. C

Explanation.

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".

Choice C is the correct answer.

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

A. Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

B. Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

C. Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

D. Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

Answer. D

Explanation.

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

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev
Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev
log827 = 2 * log16
log916 = b 
log169 = 1/b

log827 = 2/b

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

The question is "find log8108."

Hence, the answer is Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

Choice D is the correct answer. 

Question 5: Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev 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

Answer. A

Explanation.

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

log3x = y

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

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".

Choice A is the correct answer.

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

A. Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev
B. (a + b) * 2ab

C. Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev
D. Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

Answer. A

Explanation.

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

logx20 = 1/b
Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev
Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev
Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev
Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev
Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

The question is "What is logx10?"

Hence, the answer is Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

Choice A is the correct answer. 

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

A. 34

B. 31/8

C. 31/4

D. 31/3

Answer. C

Explanation.

log3x + logx3 = 17/4

Let y = log3x
We know that logx3 = Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

Hence logx3 = 1/y

Thus the equation can be written asLogarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

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".

Choice A is the correct answer.

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

A. 4

B. 3

C. 31/2

D. 31/16

Answer. B

Explanation.

logxy + logyx2 = 3
Let a = logxy 

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

logyx2 = 2logyx
We know that logyx =Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

Hence form above logyx = 1/a

Now rewritting the equation logxy + logyx2 = 3 

Using a we get Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

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".

Choice B is the correct answer.

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

Answer. B

Explanation.

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:- 

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

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".

Choice B is the correct answer.

Question 10: If 33 + 6 + 9 + ……… 3x =Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev what is the value of x? 

A. 3

B. 6

C. 7

D. 11

Answer. D

Explanation.

First of all, let us define the xth term. 

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

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 Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev must be some form of 3a. 

With little hit and trial, you may find Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

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

 Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

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".

Choice D is the correct answer.

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

Answer. A

Explanation

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

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

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".

Choice A is the correct answer.

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

A. 0

B. 3

C. 0 and 3

D. none of these

Answer. D

Explanation.

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 

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

⇒ 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".

Choice D is the correct answer.

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

A. 7

B. 6

C. 9

D. 12

Answer. A

Explanation.

Take right side expression, 

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

= (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

Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev

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".

Choice A is the correct answer.


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