Table of contents 
Logarithm 
Properties of Logarithms 
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1. log a (xy) = log_{a} x + log_{a} y
3. log_{x} x = 1
4. log_{a} 1 = 0
5. log_{a} (x^{n}) = n(log_{a} x)
1. Characteristic
The internal part of the logarithm of a number is called its characteristic.
Case I: When the number is greater than 1.
Case II: When the number is less than 1.
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 log_{2}X + log_{4}X = log_{0.25}√6 and x > 0, then x is:
A. 6^{1/6}
B. 6^{1/6}
C. 3^{1/3}
D. 6^{1/3}
Correct Answer is Option (A).
Question 2: log_{9} (3log_{2} (1 + log_{3} (1 + 2log_{2}x))) = 1/2. Find x.
A. 4
B. 1/2
C. 1
D. 2
Correct Answer is Option (D).
log_{9} (3log_{2} (1 + log_{3} (1 + 2log_{2}x)) = 1/2
3log_{2}(1 + log_{3}(1 + 2log_{2}x)) = 9^{1/2} = 3
log_{2}(1 + log_{3}(1 + 2log_{2}x) = 1
1 + log_{3}(1 + 2log_{2}x) = 2
log_{3}(1 + 2log_{2}x) = 1
1 + 2log_{2}x = 3
2log_{2}x = 2
log_{2}x = 1
x = 2
The question is "Find x."
Question 3: If 2^{2x+4} – 17 × 2^{x+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).
2^{x+4} – 17 * 2^{x+1} = – 4
=> 2^{x+1} = y
2^{2x+2} = y^{2}
2^{2}(2^{2x+2}) – 17 * 2^{x+1} = –4
4y^{2} – 17y + 4 = 0
4y^{2} – 16y – y + y = 0
4y (y – 4) – 1 (y – 4) = 0
y = 1/4 or 4
2^{x+1} = 1/4 or 4
⇒ x + 1 = 2 or – 2
x = 1 or – 3
The question is "which of the following is true?"
Question 4: If log_{12}27 = a, log_{9}16 = b, find log_{8}108
A.
B.
C.
D.
Correct Answer is Option (D).
log_{8}108 = log_{8}(4 * 27)
log_{8}108 = log_{8}4 + log_{8}27
⇒ log_{8}4 = 2/3
log_{8}27 = 2 * log_{16}9
log_{9}16 = b
log_{16}9 = 1/b
log_{8}27 = 2/b
The question is "find log_{8}108."
Question 5: 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).
log_{3}x = y
⇒
y ∈ (3, 5)
3 < log_{3}x < 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."
Question 6: log_{5}x = a (This should be read as log X to the base 5 equals a) log_{20}x = b. What is log_{x}10?
A.
B. (a + b) * 2ab
C.
D.
Correct Answer is Option (A).
Given, log_{5}x = a
log_{20}x = b
log_{x}5 = 1/a
log_{x}20 = 1/b
⇒
The question is "What is log_{x}10?"
Question 7: log_{3}x + log_{x}3 = 17/4. Find x.
A. 34
B. 3^{1/8}
C. 3^{1/4}
D. 3^{1/3}
Correct Answer is Option (C).
log_{3}x + log_{x}3 = 17/4
Let y = log_{3}x
We know that log_{x}3 =
Hence log_{x}3 = 1/y
Thus the equation can be written as
4y^{2} + 4 = 17y
4y^{2} + 4  17y = 0
Solving the above equation we get y = 4 or 1/4
If y = 4
log_{3}x = 4
Then x = 3^{4}
If y = 1/4
log_{3}x = 1/4
Then x = 3^{1/4}
The question is "Find x."
Question 8: log_{x}y + log_{y}x^{2} = 3. Find log_{x}y^{3}.
A. 4
B. 3
C. 3^{1/2}
D. 3^{1/16}
Correct Answer is Option (B).
log_{x}y + log_{y}x^{2} = 3
Let a = log_{x}y
log_{y}x^{2} = 2log_{y}x
We know that log_{y}x =
Hence form above log_{y}x = 1/a
Now rewritting the equation log_{x}y + log_{y}x^{2} = 3
Using a we get
i.e., a^{2}  3a + 2 = 0
Solving we get a = 2 or 1
If a = 2, Then log_{x}y = 2 and log_{y}x^{3} = 3
log_{x}y = 3 * 2 = 6
Or
If a = 1, Then log_{x}y = 1 and log_{y}x^{3} = 3
log_{x}y = 3 * 1 = 3
The question is "Find log_{x}y^{3}."
Question 9: log_{2} 4 * log_{4} 8 * log_{8} 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)} 2.2^{n}
Given,
log_{2} 4 * log_{4} 8 * log_{8} 16 * ……………log_{(2n)} 2.2n = 49
Whenever solving a logarithm equation, generally one should approach towards making the base same.
Making the base 2:
log_{(2n)} 2.2^{n} = 49
log_{(2n)} 2 + log_{(2n)} 2^{n} = 49
1 + n = 49
n = 48
The question is "what is the value of n?"
Question 10: If 3^{3 + 6 + 9 + ……… 3x }= 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 x^{th} term.
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 3^{a}, so must be some form of 3^{a}.
With little hit and trial, you may find
3^{3(1 + 2 + 3 + ...X)} = 3^{ 3 * 66}
⇒ 3^{3} * 3^{x(x+1)/2} = 3^{3*66}
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?"
Question 11: x, y, z are 3 integers in a geometric sequence such that y  x is a perfect cube. Given, log_{36}x^{2} + log_{6}√y + log_{216}y^{1/2}z = 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:
⇒ log_{62}x^{2} + log_{6}y^{1/2} + 3log_{63}y^{1/2}z = 6
log_{6}x + log_{6}y^{1/2}y^{1/2}z = 6
log_{6}xyz = 6
xyz = 6^{6}
Given x,y,z is in G.P. Let x = a, y = ab, z = ab^{2}
⇒ xyz = a^{3}b^{3} = (ab)^{3}
(ab)^{3} = (6^{2})^{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 yx is a perfect cube
⇒ aba is perfect cube
⇒ a(b1) 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."
Question 12: 10^{log(3  10logy)} = log_{2}(9  2^{y}), 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
10^{log(3y)} = log_{2}(9  2^{y}) (y > 0)…………………………………(1)
3  y = log_{2}(9  2^{y}) (Therefore, 3  y > 0 =) (y < 3)) ……………………… (2)
2^{3y} = 9  2^{y}
2^{y} = t
⇒ 8 = 9t –t^{2}
⇒ t^{2}  9t + 8 = 0
⇒ t^{2}  t  8t  8 = 0
⇒ t(t  1)  8(t  1) = 0
⇒ t = 1, 8
Therefore, 2^{y} = 1 and 2^{y} = 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."
Question 13: 4^{6+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,
= (4^{2})^{84}
= 4^{168}
Take left side expression
4^{6+12+18+24+…+6x} = 4^{6(1+2+3+4+x)}
= 4^{6} * 4^{(1+2+3+4+…+x)}
= 4^{6} * 4^{x(x+1)/2} (using the formula for sum of natural numbers from 1 to x)
Equating left and right side expresssions, we get 4^{6} * 4^{x(x+1)/2} = 4^{168}
Or 4^{6} * 4^{x(x+1)/2} = 4^{6*28}
⇒
or x (x + 1) = 56
Solving for x we get, x = 7
The question is "what is the value of x?"
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