Surds, Indices and Logarithm

PrepBank · পাতা ১ / ৫ · ১–১০০ / ৪৭১

১.

If ‘x’ is an integer then solve (log₂x)² - log₂x⁴ - 32 = 0.

ক
125
খ
256
গ
375
ঘ
373
ঙ
None of these

ব্যাখ্যা

Question: If ‘x’ is an integer then solve (log₂x)² - log₂x⁴ - 32 = 0.

Solution:
We have (log₂x)² - log₂x⁴ - 32 = 0.
⇒ (log₂x)² - 4log₂x - 32 = 0 ......(1)
Let log₂x = y
(i) ⇒ y² - 4y - 32 = 0
⇒ y² - 8y + 4y - 32 = 0
⇒ y (y - 8) + 4(y - 8) = 0
⇒ (y - 8) (y + 4) = 0
⇒ y = 8, - 4
⇒ log₂x = 8 or log₂x = - 4
⇒ x = 2⁸ = 256 or x = 2^{- 4} = 1/16
Since ‘x’ is an integer so x = 256.

২.

ক
ক) 7
খ
খ) 9
গ
গ) 21
ঘ
ঘ) 49

ব্যাখ্যা

Question:

Solution:

৩.

If 3^{x +1} = 243, what is the value of 2^{2x - 7}?

ক
ক) 1
খ
খ) 2
গ
গ) 5
ঘ
ঘ) 8

ব্যাখ্যা

Question: If 3^{x +1} = 243, what is the value of 2^{2x - 7}?

Solution:
3^{x +1} = 243
⇒ 3^{x + 1} = 3⁵
⇒ x + 1 = 5
⇒ x = 5 - 1
⇒ x = 4

2^{2x - 7} = 2^{2 × 4 - 7} = 2^{8 - 7} = 2¹ = 2

৪.

The solution to log₂x = 5 is:

ক
10
খ
25
গ
15
ঘ
32

ব্যাখ্যা

Question: The solution to log₂x = 5 is:

Solution:
log₂x = 5
⇒ x = 2⁵
∴ x = 32

৫.

If m is an integer such that (- 2)^2m = 2^{9 - m}, then m =?

ক
1
খ
2
গ
3
ঘ
4
ঙ
6

ব্যাখ্যা

Question: If m is an integer such that (- 2)^2m = 2^{9 - m}, then m =?

Solution:
First of all, since m is an integer, then 2m = even, and therefore (- 2)^2m = 2^2m

So, we'd have:
2^2m = 2^{9 - m}
⇒ 2m = 9 - m
⇒ 3m = 9
∴ m = 3

৬.

(5³ + 5³ + 5³ + 5³) = ?

ক
5⁴
খ
5⁶
গ
5⁸
ঘ
5¹²
ঙ
None of these

ব্যাখ্যা

Question: (5³ + 5³ + 5³ + 5³) = ?

Solution:
5³ + 5³ + 5³ + 5³
= 5³ (1 + 1 + 1 + 1)
= 5³ · 4
= 125 · 4
= 500

৭.

Find the value of i^-42

ক
ক) 1
খ
খ) -1
গ
গ) i
ঘ
ঘ) -i

ব্যাখ্যা

Question: Find the value of i^-42

Solution:
i^-42
= 1 / i⁴²
= 1 / {(i²)²¹}
= 1 / {(-1)²¹} [i² = -1]
= 1 / (-1)
= - 1

৮.

4 log 2 + 2 log 3 - 1/2 log 81 = ?

ক
log 7.2
খ
log 12
গ
log 9
ঘ
log 16

ব্যাখ্যা

Question: 4 log 2 + 2 log 3 - 1/2 log 81 = ?

Solution:
4 log 2 + 2 log 3 - 1/2 log 81
= log 2⁴ + log 3² - log(81)^1/2
= log 16 + log 9 - log √81
= log 16 + log 9 - log 9
= log(16 × 9) - log 9
= log 144 - log 9
= log(144/9)
= log 16

৯.

If √2ⁿ = 64 , what will be the value of n?

ক
ক) 12
খ
খ) 8
গ
গ) 4
ঘ
ঘ) 16

ব্যাখ্যা

√(2ⁿ) = 64
⇒ √(2ⁿ) = 2⁶
⇒ 2ⁿ = (2⁶)²
⇒ 2ⁿ= 2¹²
⇒ n = 12

১০.

Find the value of n, if 64^{n - (2/3)} = 256

ক
2
খ
- 1
গ
0
ঘ
6

ব্যাখ্যা

Question: Find the value of n, if 64^{n - (2/3)} = 256

Solution:
64^{n - (2/3)} = 256
⇒ (4³)^{n - (2/3)} = 4⁴
⇒ 4^{(3n - 2)} = 4⁴
⇒ 3n - 2 = 4
⇒ 3n = 4 + 2
⇒ 3n = 6
∴ n = 2

১১.

There are 10 true false questions in an examination. These questions can be answered in -

ক
ক) 20 ways
খ
খ) 100 ways
গ
গ) 210 ways
ঘ
ঘ) 1024 ways

ব্যাখ্যা

Each question can be answered in two ways: True or False
So, These questions can be answered in = 2¹⁰ ways = 1024 ways

১২.

x^1/2/27 = 12/x^3/2, What is the value of x is-

ক
15
খ
4
গ
25
ঘ
18

ব্যাখ্যা

Question: x^1/2/27 = 12/x^3/2, What is the value of x is-

Solution:
Given that,
⇒ x^1/2/27 = 12/x^3/2
⇒ x^1/2 × x^3/2 = 27 × 12
⇒ x^{{(1/2) + (3/2)}} = 324
⇒ x^4/2 = 324
⇒ x² = 324
⇒ x = √324
∴ x = 18

১৩.

4, -8, 16, -32, 64, (....)

ক
ক) 128
খ
খ) -128
গ
গ) 192
ঘ
ঘ) -192

ব্যাখ্যা

Each number is the proceeding number multiplied by -2. So, the required number is -128.

১৪.

Solve for x, log₃(2x + 1) = 4

ক
45
খ
37
গ
81
ঘ
40

ব্যাখ্যা

Question: Solve for x, log₃(2x + 1) = 4

Solution:
Given that,
⇒ log₃(2x + 1) = 4

⇒ 2x + 1 = 3⁴
⇒ 2x + 1 = 81
⇒ 2x = 81 - 1
⇒ 2x = 80

১৫.

If 4^{x + y} = 1 and 4^{x - y} = 4, then the values of x and y respectively are-

ক
- 1/2 and 1/2
খ
- 1/2 and - 1/2
গ
1/2 and - 1/2
ঘ
1/2 and 1/2

ব্যাখ্যা

Question: If 4^{x + y} = 1 and 4^{x - y} = 4, then the values of x and y respectively are-

Solution:
4^{x + y} = 1
⇒ 4^{x + y} = 4⁰
⇒ x + y = 0 ..............(1)

4^{x - y} = 4
⇒ 4^{x - y} = 4¹
⇒ x - y = 1 ..................(2)

x + y + x - y = 1
⇒ 2x = 1
∴ x = 1/2

1/2 - y = 1
⇒ y = (1/2) - 1
y = (1 - 2)/2
= - 1/2

১৬.

log₃(x⁴ - x³) - log₃(x - 1) = 3 then x is equal to-

ক
1
খ
3
গ
2
ঘ
4

ব্যাখ্যা

Question: log₃(x⁴ - x³) - log₃(x - 1) = 3 then x is equal to-

Solution:
Given that,
⇒ log₃(x⁴ - x³) - log₃(x - 1) = 3
⇒ log₃{(x⁴ - x³)/(x - 1)} = 3
⇒ log₃{x³(x - 1)/(x - 1)} = 3
⇒ log₃x³ = 3
⇒ 3log₃x = 3
⇒ log₃x = 1
⇒ x = 3¹
∴ x = 3

১৭.

What is the value of x in the equation log_x(1/81) = 4

ক
ক) 1/3
খ
খ) 1/2
গ
গ) 3
ঘ
ঘ) 2

ব্যাখ্যা

log_x(1/81) = 4
⇒ x⁴ = 1/81
⇒ x⁴ = (1/3)⁴
∴ x = 1/3

১৮.

3³⁰ + 3³⁰ + 3³⁰ = ?

ক
3⁶⁰
খ
3³³
গ
3³¹
ঘ
9³⁰

ব্যাখ্যা

Question: 3³⁰ + 3³⁰ + 3³⁰ = ?

Solution:
3³⁰ + 3³⁰ + 3³⁰
= 3 × 3³⁰
= 3^{1 + 30}
= 3³¹

১৯.

If an exponent or index has base 100 and power zero, then which of the following will be its value?

ক
100
খ
5
গ
10
ঘ
1

ব্যাখ্যা

Question: If an exponent or index has base 100 and power zero, then which of the following will be its value?
(Officer Cash 2022 অনুযায়ী)

Solution:
x⁰ = 1 (for any non-zero base a)

Now,
= 100⁰
= 1

২০.

If log₂[log₃(log₂x)] = 1, then x is equal to = ?

ক
0
খ
12
গ
128
ঘ
512

ব্যাখ্যা

Question: If log₂[log₃(log₂x)] = 1, then x is equal to = ?

Solution:
Given that,
log₂[ log₃( log₂x)]=1
⇒ log₃(log₂x) = 2¹ = 2
⇒ log₂x = 3² = 9
⇒ x = 2⁹ = 512
∴ x = 512

২১.

If log₃(x⁴ - x³) - log₃(x - 1) = 3 then x is equal to?

ক
6
খ
9
গ
1
ঘ
3

ব্যাখ্যা

Question: If log₃(x⁴ - x³) - log₃(x - 1) = 3 then x is equal to?

Solution:
Given that,
log₃(x⁴ - x³) - log₃(x - 1) = 3
⇒ log₃[(x⁴ - x³)/(x - 1)] = 3
⇒ log₃[x³(x - 1)/(x - 1)] = 3
⇒ log₃x³ = 3
⇒ 3log₃x = 3
⇒ log₃x = 3/3 = 1
⇒ log₃x = 1
⇒ x = 3¹
∴ x = 3

২২.

log₁₀2500−log₁₀25

ক
ক) 4
খ
খ) 5
গ
গ) 2
ঘ
ঘ) 1

ব্যাখ্যা

log₁₀2500−log₁₀25
=log₁₀(2500 / 25)
=log₁₀100
=log₁₀10²=2log₁₀10
=2

২৩.

ক
8/15
খ
31/7
গ
12/13
ঘ
17/11

ব্যাখ্যা

Question:

Solution:

২৪.

If log_m (1/√32) = - 5/2 what is the value of m?

ক
2
খ
- 3/4
গ
3
ঘ
5

ব্যাখ্যা

Question: If log_m (1/√32) = - 5/2 what is the value of m?

Solution:
দেওয়া আছে,
log_m (1/√32) = - 5/2
⇒ m^{- 5/2} = 1/√32 [যেহেতু log_aM = x হলে, a^x = M]
⇒ m^{- 5/2} = 1/(32^1/2)
⇒ m^{- 5/2} = 32^{- 1/2}⇒ m^{- 5/2} = (2⁵)^{- 1/2}⇒ m^{- 5/2} = 2^{- 5/2}∴ m = 2

২৫.

If log₇(x² - x) − log₇(x - 1) = 2, then the value of x is -

ক
49
খ
42
গ
36
ঘ
30

ব্যাখ্যা

Question: If log₇(x² - x) − log₇(x - 1) = 2, then the value of x is -

Solution:
log₇(x² - x) − log₇(x - 1) = 2
⇒ log₇{(x² - x)/(x - 1)} = 2
⇒ log₇{x(x - 1)/(x - 1)} = 2
⇒ log₇x = 2
⇒ x = 7²
∴ x = 49

২৬.

If 3^2x . 9^{x + 1} = 27^{x - 1}, what is the value of x?

ক
- 5
খ
-1
গ
3
ঘ
5

ব্যাখ্যা

Question: If 3^2x . 9^{x + 1} = 27^{x - 1}, what is the value of x?

Solution:
3^2x . 9^{x + 1} = 27^{x - 1}
⇒ 3^2x . (3²)^{x + 1} = (3³)^{x - 1}
⇒ 3^2x . 3^{2(x + 1)} = 3^{3(x - 1)}
⇒ 3^2x . 3^{2x + 2} = 3^{3x - 3}
⇒ 3^{2x + 2x + 2} = 3^{3x - 3}
⇒ 3^{4x + 2} = 3^{3x - 3}
⇒ 4x + 2 = 3x - 3
⇒ 4x - 3x = - 3 - 2
∴ x = - 5

২৭.

ক
7
খ
14
গ
49
ঘ
50

ব্যাখ্যা

Question:
Solution:

২৮.

If log₂(a) = 3 and log₂(b) = 4, what is the value of log₂(ab) =?

ক
12
খ
28
গ
14
ঘ
7

ব্যাখ্যা

Question: If log₂(a) = 3 and log₂(b) = 4, what is the value of log₂(ab) =?

Solution:
given that,
log₂(a) = 3 and log₂(b) = 4

Now,
log₂(ab) = log₂(a) + log₂(b) ;[log_b(mn) = log_b(m) + log_b(n)]
= 3 + 4
= 7

২৯.

If 8 × 2^x = (1/16), what is the value of x?

ক
ক) 1
খ
খ) - 7
গ
গ) 7
ঘ
ঘ) -8

ব্যাখ্যা

প্রশ্ন: If 8 × 2^x = (1/16), what is the value of x?

সমাধান:
8 × 2^x= (1/16)
⇒ 2^x = (1/16 × 8)
⇒ 2^x= 1/(2⁴ × 2³)
⇒ 2^x = 1/2⁷
⇒ 2^x= 2 ^{- 7}
∴ X = -7

৩০.

If 4^x-1 + 4^x+1 = 4352, then find the value of x.

ক
ক) 0
খ
খ) 1
গ
গ) 2
ঘ
ঘ) 5

ব্যাখ্যা

Question: If 4^x-1 + 4^x+1 = 4352, then find the value of x.

Solution:
Given that,
4^x-1 + 4^x+1 = 4352
⇒ 4^x/4 + 4^x × 4 = 4352
⇒ 4^x(1/4 + 4) = 4532
⇒ 4^x × (17/4) = 4532
⇒ 4^x = (4532 × 4)/17
⇒ 4^x = 1024
⇒ 4^x = 4⁵
∴ x = 5

৩১.

(289)^0.17 × (17)^0.16= ?

ক
4
খ
√7
গ
√17
ঘ
√19

ব্যাখ্যা

Question: (289)^0.17 × (17)^0.16= ?

Solution:
Given that,
(289)^0.17 × (17)^0.16
= (17²)^0.17 × (17)^0.16
= (17)^0.34 ×(17)^0.16
= (17)^{0.34 + 0.16}
= (17)^0.50
= (17)^1/2
= √17

৩২.

If n = 3⁸ - 2⁸, which of the following is not a factor of n?

ক
ক) 97
খ
খ) 65
গ
গ) 35
ঘ
ঘ) 13

ব্যাখ্যা

n = 3⁸ - 2⁸
= (3⁴)² - (2⁴)²
= (3⁴ + 2⁴)(3⁴ - 2⁴)
= (81 + 16)(81 - 16)
= 97 × 65
= 97 × 13 × 5

So, we can see that here 35 can't be a factor of n

৩৩.

If log_x32 = 5, the value of x is

ক
2
খ
3
গ
4
ঘ
5

ব্যাখ্যা

log_x32 = 5
or, x⁵ = 32
or, x⁵ = 2⁵
or, x = 2

৩৪.

If 2^{x - 6} = 1/32, the value of x/2 is:

ক
1/3
খ
1/4
গ
2
ঘ
1/2
ঙ
1

ব্যাখ্যা

Question: If 2^{x - 6} = 1/32, the value of x/2 is:

Solution:

৩৫.

If log_a3 = x and log_a5 = y, then log_a75 = ?

ক
ক) x + y
খ
খ) 3x + 2y
গ
গ) x + 2y
ঘ
ঘ) 2x + y

ব্যাখ্যা

Question: If log_a3 = x and log_a5 = y, then log_a75 = ?

Solution:
log_a3 = x
log_a5 = y

log_a75 = log_a(3 × 5²)
= log_a3 + log_a5²
= log_a3 + 2log_a5
= x + 2y

৩৬.

(125) ^{- 2/3}× (1/5)^{- 2} is equal to ?

ক
ক) 1
খ
খ) 5
গ
গ) 1/25
ঘ
ঘ) 1/5

ব্যাখ্যা

Question: (125) ^{- 2/3}× (1/5)^{- 2} is equal to ?

Solution:
(125) - 2/3 × (1/5)- 2
=(5³)^{- 2/3}× (1/5)^{- 2}
= 5^-2 × (1/5)^{- 2}
= (1/5)² × (1/5)^{- 2}
= (1/5)^{2 - 2}
= (1/5)⁰
= 1

৩৭.

If log_X(81/16) = -4, what is the value of X?

ক
2/3
খ
3/2
গ
4/3
ঘ
3/4

ব্যাখ্যা

Question: If log_X(81/16) = -4, what is the value of X?

Solution:
By definition of logarithm:
log_X(y) = n
⇒ Xⁿ = y

Here,
log_X(81/16) = - 4
⇒ X^-4 = 81/16
⇒ X⁴ = 16/81
⇒ X = ⁴√(16/81)
⇒ X = 2/3

৩৮.

Find the value of

ক
9
খ
6
গ
12
ঘ
7

ব্যাখ্যা

Question: Find the value of .

Solution:

৩৯.

3²⁰ + 3²⁰ + 3²⁰ = ?

ক
3²¹
খ
3²³
গ
3⁶⁰
ঘ
9²⁰

ব্যাখ্যা

Question: 3²⁰ + 3²⁰ + 3²⁰ = ?

Solution:
3²⁰ + 3²⁰ + 3²⁰
= 3 × 3²⁰
= 3¹^{+ 20}
= 3²¹

৪০.

ক
32/143
খ
16/81
গ
4/9
ঘ
3/2

ব্যাখ্যা

Question:

Solution:

৪১.

log₂√10 - log₂√(5/2) = ?

ক
2/5
খ
0
গ
1
ঘ
√2/5

ব্যাখ্যা

Question: log₂√10 - log₂√(5/2) = ?

Solution:
log₂√10 - log√(5/2)
= log₂(10)^(1/2) - log₂(5/2)^(1/2)
= (1/2)log₂(10) - (1/2)log₂(5/2)
= (1/2)[log₂(5 × 2) - log₂(5/2)]
= (1/2)[log₂5 + log₂2 - (log₂5 - log₂2)]
= (1/2)[log₂5 + 1 - log₂(5) + 1]
= (1/2) × 2
= 1

৪২.

If 2^{m + n} = √2, 81^{m - n} = 3 then what is the value of n?

ক
ক) 3/8
খ
খ) 1/4
গ
গ) 1/8
ঘ
ঘ) 5/8

ব্যাখ্যা

Question: If 2^{m + n} = √2, 81^{m - n} = 3 then what is the value of n?

Solution:
2 ^{m + n} = √2
⇒ (√2)^{2(m + n)} = √2
⇒ m + n = 1/2

81 ^{m - n} = 3
⇒ 3^{4(m - n)} = 3
⇒ m - n = 1/4

m + n -m + n = (1/2) - (1/4)
⇒ 2n = (2 - 1)/4
⇒ 2n = 1/4
∴ n = 1/8

৪৩.

(3^x + 3^x + 3^x)/3^x = ?

ক
ক) 3^x
খ
খ) 3^{x + 1}
গ
গ) 1
ঘ
ঘ) 3

ব্যাখ্যা

Question: (3^x + 3^x + 3^x)/3^x = ?

Solution:
(3^x + 3^x + 3^x)3^x
= {3^x(1 + 1 + 1)}/3x
= (3^x . 3)/3x
= 3^{(x + 1)}/3^x
= 3^{x + 1 - x}
= 3¹
= 3

৪৪.

What will come in the place of question mark (?) in the following question?

ক
ক) 3
খ
খ) 5
গ
গ) 2
ঘ
ঘ) 9

ব্যাখ্যা

? = {(243)²}^1/5
? = (243)^2/5
? = (3⁵)^2/5
? = 3²
? = 9

৪৫.

যদি log₂₇x + log₂₇(1/6) = 1/3 হয়, তবে x এর মান কত?

ক
18
খ
12
গ
9
ঘ
24

ব্যাখ্যা

প্রশ্ন: যদি log₂₇x + log₂₇(1/6) = 1/3 হয়, তবে x এর মান কত?

সমাধান:
log₂₇x + log₂₇(1/6) = 1/3
⇒ log₂₇{x × (1/6)} = 1/3
⇒ log₂₇(x/6) = 1/3
⇒ 27^1/3 = x/6
⇒ (3³)^1/3 = x/6
⇒ 3 = x/6
∴ x = 18

৪৬.

If 3^x- 3^x-1= 18, the value of x^x is

ক
ক) 3
খ
খ) 8
গ
গ) 27
ঘ
ঘ) 216

ব্যাখ্যা

Question: If 3^x - 3^{x - 1} = 18, the value of x^x is-

Solution:
3^x - 3^{x - 1} = 18
3^x - 3^x.3- 1 = 18
3^x - 3^x/3= 18
3^x(1 - 1/3) = 18
3^x (3 - 1)/3 = 18
3^x (2/3) = 18
3^x = (18 × 3)/2
3^x = 27
3^x = 3³
x = 3

x^x = 3³
= 27

৪৭.

(০.২)^২ ÷ (০.১)^৩ = কত?

ক
ক) ৩০
খ
খ) ৪০
গ
গ) ৪৪
ঘ
ঘ) ৪২

ব্যাখ্যা

প্রশ্ন: (০.২)^২ ÷ (০.১)^৩ = কত?

সমাধান:
(০.২)^২ ÷ (০.১)^৩
= (০.২ × ০.২) ÷ (০.১ × ০.১ × ০.১)
= ০.০৪ ÷ ০.০০১
= ৪০

৪৮.

(256)^0.16 × (256)^0.09 = ?

ক
4
খ
5/8
গ
1/4
ঘ
25

ব্যাখ্যা

Question: (256)^0.16 × (256)^0.09 = ?

Solution:
(256)^{0.16 + 0.09}= (256)^0.25= (256)^25/100
= (256)^1/4= (4⁴)^1/4 (because 256 = 4⁴)
= 4^{4 × 1/4}= 4¹= 4

৪৯.

If 7^{(x + 2)} = 49^{(3x - 4)} then the value of x = ?

ক
0
খ
2
গ
5
ঘ
7

ব্যাখ্যা

Question: If 7^{(x + 2)} = 49^{(3x - 4)} then the value of x = ?

Solution:
7^{(x + 2)} = 49^{(3x - 4)}
⇒ 7^{(x + 2)} = (7²)^{(3x - 4)}
⇒ 7^{(x + 2)} = 7^{(6x - 8)}
⇒ x + 2 = 6x - 8
⇒ 5x = 10
∴ x = 2

৫০.

9³ × (81)² ÷ (27)³ = (3)^?

ক
ক) 3
খ
খ) 5
গ
গ) 4
ঘ
ঘ) 6

ব্যাখ্যা

Question: 9³ × (81)² ÷ (27)³ = (3)^?

Solution:
ধরি,
9³ × (81)² ÷ (27)³ = (3)^x
⇒ (3²)³ × (3⁴)² ÷ (3³)³ = 3^x
⇒ (3⁶× 3⁸) ÷ 3⁹ = 3^x
⇒ 3^{6 + 8} ÷ 3⁹ = 3^x
⇒ 3¹⁴ ÷ 3⁹ = 3^x
⇒ 3^{14 - 9} = 3^x
⇒ 3^x = 3⁵
∴ x = 5

৫১.

What is the greatest prime factor of (2⁴)² - 1?

ক
ক) 3
খ
খ) 5
গ
গ) 11
ঘ
ঘ) 17

ব্যাখ্যা

এখানে
(2⁴)² - 1 = (2⁴)² - 1²
= (2⁴ + 1)(2⁴ - 1)
  = (16 + 1)(16 - 1)
  = 17 × 15
  = 17 × 3 × 5

সুতরাং, বৃহত্তম মৌলিক উৎপাদক হলো 17

৫২.

If a^2x+2=1, where a is a positive real number other than 1, then x = ?

ক
ক) -2
খ
খ) -1
গ
গ) 0
ঘ
ঘ) 1

ব্যাখ্যা

Given that, a^2x+2=1
=> a^2x+2=a⁰=> 2x+2=0
=> x=−2/ 2 =−1

৫৩.

If (1/2)log (11 + 4√7) = log (2 + x), what is the value of x?

ক
√7
খ
11
গ
4
ঘ
2

ব্যাখ্যা

Question: If (1/2)log (11 + 4√7) = log (2 + x), what is the value of x?

Solution:
We have (1/2)log (11 + 4√7) = log (2 + x)
Now, we can write it as (1/2)log (7 + 4 + 4√7) = log (2 + x)
⇒ (1/2)log {2² + (√7)² + 2.2.√7} = log (2 + x)
⇒ (1/2)log(2 + √7)² = log(2 + x)
⇒ 2.(1/2)log (2 +√7) = log (2 + x)
⇒ log (2 +√7) = log (2 + x)
Both side Log will be canceled out
Now, 2 + √7 = 2 + x
Therefore, x = 2 + √7 - 2 = √7

৫৪.

If 2^{x + 1} = 16, then x = ?

ক
2
খ
3
গ
4
ঘ
5

ব্যাখ্যা

Question: If 2^{x + 1} = 16, then x = ?

Solution:
Given that,
2^{x + 1} = 16
⇒ 2^{x + 1} = 2⁴
⇒ x + 1 = 4
⇒ x = 4 - 1
∴ x = 3

So the value of x is 3

৫৫.

If x = 42^0.50, y = 42^0.25 and x^z = y⁴ , then the value of z is-

ক
1/2
খ
3
গ
3/2
ঘ
2

ব্যাখ্যা

Question: If x = 42^0.50, y = 42^0.25 and x^z = y⁴ , then the value of z is-

Solution:
Given that,
x = 42^0.50, y = 42^0.25

Now,
⇒ x^z = y⁴
⇒ (42^0.50)^z = (42^0.25)⁴
⇒ 42^0.50z = 42¹
⇒ 0.50z = 1
⇒ z = 1/0.50
∴ z = 2

৫৬.

If log27 = 1.431, then the value of log9 is-

ক
0.934
খ
0.945
গ
0.954
ঘ
0.958

ব্যাখ্যা

Question: If log27 = 1.431, then the value of log9 is-

Solution:

log27 = 1.431

⇒ log(3³) = 1.431

⇒ 3log3 = 1.431

⇒ log3 = 0.477

∴ log 9 = log(3²) = 2log3 = (2 × 0.477) = 0.954

৫৭.

What is the value of x if, 8^{2x - 1} = 16^{x + 1}?

ক
5/2
খ
7/2
গ
9/2
ঘ
3

ব্যাখ্যা

Question: What is the value of x if, 8^{2x - 1} = 16^{x + 1}?

Solution:
8^{2x - 1} = 16^{x + 1}
or, 2^{3(2x - 1)} = 2^{4(x + 1)}
or, 6x - 3 = 4x + 4
or, 2x = 7
∴x = 7/2

৫৮.

If log_x(1/9) = - 2, then x = ?

ক
ক) - 1/3
খ
খ) 1/3
গ
গ) - 3
ঘ
ঘ) 3

ব্যাখ্যা

দেওয়া আছে,
log_x(1/9) = - 2
বা, 1/9 = x - 2
বা, 1/3² = 1/x²
বা, 1/x = 1/3
বা, x = 3

৫৯.

If log_a324= 4, what is the value of the base?

ক
4
খ
2√3
গ
1/2√3
ঘ
3√2

ব্যাখ্যা

Question: If log_a324= 4, what is the value of the base?

Solution:
log_a324 = 4
⇒ a⁴= 324
⇒ a⁴= 4 × 81
⇒ a⁴=2² × 3⁴
⇒ a⁴= ((√2)²)² × 3⁴
⇒ a⁴ = (√2)⁴ × 3⁴
⇒ a⁴ = (3√2)⁴
∴ a = 3√2

৬০.

If 2^{2x - 1}= 1/(8^{x - 3}), then the vale of x is-

ক
ক) - 2
খ
খ) 2
গ
গ) 0
ঘ
ঘ) 5

ব্যাখ্যা

2^{2x - 1}= 1/(8^{x - 3})
2^{2x - 1} = 1/{(2³)^{x - 3}}
2^{2x - 1} = 1/2^{3x - 9}
2^{2x - 1} = 2^{- (3x - 9)}
2^{2x - 1} =2 ^{- 3x + 9}
2x - 1 = - 3x + 9
2x + 3x = 9 + 1
5x = 10
x = 10/5
x = 2

৬১.

If 2^{x - 1} + 2^{x + 1} = 320, then x is equal to-

ক
5
খ
6
গ
7
ঘ
8
ঙ
4

ব্যাখ্যা

Question: If 2^{x - 1} + 2^{x + 1} = 320, then x is equal to-

Solution:

∴ x - 6 = 1
⇒ x = 7

৬২.

√(5/7)^{6x - 4} = 2401/875 Find value of x

ক
3
খ
5
গ
- 1/2
ঘ
- 1/3

ব্যাখ্যা

Question: √(5/7)^{6x - 4} = 2401/875 Find value of x

Solution:
√(5/7)^{6x - 4} = 2401/875
⇒ (5/7)^{3x - 2} = 7⁴/(7 × 5³) = (7/5)³
⇒ (5/7)^{3x - 2} = (5/7)^{- 3}
⇒ 3x - 2 = - 3
⇒ 3x = - 1
∴ x = - 1/3

৬৩.

Find the value of x, if (log₁₀ 225/log₁₀15) = log₁₀ x.

ক
49
খ
100
গ
64
ঘ
25

ব্যাখ্যা

Question: Find the value of x, if (log₁₀ 225/log₁₀15) = log₁₀ x.

Solution:
⇒ log₁₀ x = (log₁₀ 225/log₁₀15)
⇒ log₁₀ x = [log₁₀(15 × 15)/log₁₀15]
⇒ log₁₀ x = log₁₀15²/log₁₀15
⇒ log₁₀ x = 2(log₁₀15/log₁₀15)
⇒ log₁₀ x = 2
⇒ x = 10²
∴ x = 100

৬৪.

a^{(x - y)(x + y)} × a^{(z - x)(z + x)} × a^{(y - z)(y + z)} = ?

ক
0
খ
1
গ
a
ঘ
1/2

ব্যাখ্যা

Question: a^{(x - y)(x + y)} × a^{(z - x)(z + x)} × a^{(y - z)(y + z)} = ?

Solution:

৬৫.

If (16)^{2x + 3} = (4)^{3x + 6}, than x = ?

ক
- 2
খ
4
গ
0
ঘ
- 3

ব্যাখ্যা

Question: If (16)^{2x + 3} = (4)^{3x + 6}, than x = ?

Sulotion:
Given that,
⇒ (16)^{2x + 3} = (4)^{3x + 6}
⇒ (2⁴)^{2x + 3}= (2²)^{3x + 6}
⇒ (2)^{8x + 12} = (2)^{6x + 12}
⇒ 8x + 12 = 6x + 12
⇒ 8x - 6x = 12 - 12
⇒ 2x = 0
∴ x = 0

৬৬.

If log_m 128 = 7, then find the value of m.

ক
2
খ
3
গ
5
ঘ
7

ব্যাখ্যা

Question: If log_m 128 = 7, then find the value of m.

Solution:
log_m 128 = 7
⇒ m⁷ = 128 [log_b A = C, then b^C = A]
⇒ m⁷ = 2⁷
∴ m = 2

৬৭.

If a^1/x = b^1/y = c^1/z and abc = 1, then find the value of x + y + z.

ক
- 1
খ
1
গ
1/2
ঘ
0

ব্যাখ্যা

Question: If a^1/x = b^1/y = c^1/z and abc = 1, then find the value of x + y + z.

Solution:
Let, a^1/x = b^1/y = c^1/z = k
a = k^x, b = k^y and c = k^z
abc = k^x × k^y × k^z = k^{(x + y + z)}Given, abc = 1
k^{(x + y + z)} = k⁰
x + y + z = 0

৬৮.

log 2 + log 4 + log 8 + ............ Find the sum of the first 19th term-

ক
55 log 2
খ
120 log 2
গ
190 log 2
ঘ
210 log 2

ব্যাখ্যা

Question: log 2 + log 4 + log 8 + ............ Find the sum of the first 19th term-

Solution:
given that,
log2 + log4 +log8 + ............
= log 2¹+ log 2² + log 2³ + ............
= log 2 + 2 log 2 + 3 log 2 + ............
= (1 + 2 + 3 + .........) × log 2

The sum of the first 19 natural numbers is given by the formula:
Sum = n(n+1)/2
where n = 19
∴ Sum = 19(19 + 1)/2
= 19 × 10
= 190

So, the sum of the first 19 terms = 190 log 2

৬৯.

If m is an integer such that (- 2)^2m = 2^{9 - m}, then what is the value of m?

ক
3
খ
- 3
গ
1/2
ঘ
4

ব্যাখ্যা

Question: If m is an integer such that (- 2)^2m = 2^{9 - m}, then what is the value of m?

Solution:
দেওয়া আছে,
(- 2)^2m = 2^{9 - m}
⇒ 2^2m = 2^{9 - m}
কোনো ঋণাত্মক সংখ্যার Power যদি জোড় সংখ্যা হয় তবে সংখ্যাটি ধনাত্মক হবে। আর যদি Power বিজোড় হয় তবে সংখ্যাটি ঋণাত্মক হবে। যেহেতু m একটি পূর্ণ সংখ্যা সেহেতু 2m একটি জোড় সংখ্যা। তাই (- 2)^2m সংখ্যাটি ধনাত্মক সংখ্যা হবে।

∴ 2m = 9 - m
⇒ 3m = 9
⇒ m = 9/3
⇒ m = 3

৭০.

If log_x(125/8) = - 3, what is the value of x?

ক
3/5
খ
2/5
গ
5/3
ঘ
3/8

ব্যাখ্যা

Question: If log_x(125/8) = - 3, what is the value of x?

Solution:
log_x(125/8) = - 3
⇒ x^{- 3} = 125/8 [log_b(a) = c ⇒ b^c = a]
⇒ x^{- 3} = 5³/2³
⇒ x^{- 3} = (5/2)³
⇒ x^{- 3} = (2/5)^{- 3}
∴ x = 2/5

৭১.

If log_a2 = a and log_a5 = b, then log_a50 =

ক
ক) a + b
খ
খ) a + b²
গ
গ) ab²
ঘ
ঘ) a + 2b

ব্যাখ্যা

Question: If log_a2 = a and log_a5 = b, then log_a50 =

Solution:
log_a2 = a
log_a5 = b

log_a50 = log_a(2 × 5²)
= log_a2 + log_a5²
= log_a2 + 2log_a5
= a + 2b

৭২.

Write in terms of indices: log₂₇81= 4/3

ক
ক) 3⁴
খ
খ) 3³
গ
গ) 27^1/3
ঘ
ঘ) 27^2/3

ব্যাখ্যা

log₂₇81 = 4/3
81 = (27)^4/3 = (3³)^4/3 = 3⁴

৭৩.

The value of log₃2 ⋅ log₄3 ⋅ log₅4 ⋅ log₆5 ⋅ log₇6 ⋅ log₈7 is-

ক
1/2
খ
2
গ
1/3
ঘ
1
ঙ
1/4

ব্যাখ্যা

Question: The value of log₃2 ⋅ log₄3 ⋅ log₅4 ⋅ log₆5 ⋅ log₇6 ⋅ log₈7 is-

Solution:
log₃2 ⋅ log₄3 ⋅ log₅4 ⋅ log₆5 ⋅ log₇6 ⋅ log₈7
= (log₃2 ⋅ log₄3) (log₅4 ⋅ log₆5) (log₇6 ⋅ log₈7)
= log₄2 · log₆4 · log₈6 [log_bM × log_ab = log_aM]
= (log₄2 · log₆4) log₈6
= log₆2 ⋅ log₈6
= log₈2
= 1/log₂8
= 1/log₂2³
= 1/(3log²2)
= 1/(3 × 1) [∵ log₂2 = 1]
= 1/3

৭৪.

ক
14
খ
6
গ
12
ঘ
49
ঙ
7

ব্যাখ্যা

Question:

Solution:

৭৫.

If x^a = y^b, then

ক
ক) logx/logy = a/b
খ
খ) logx/logy = b/a
গ
গ) logx/logy = ab
ঘ
ঘ) None of these

ব্যাখ্যা

Question: If x^a = y^b, then

Solution:
x^a = y^b
Take log on both the sides, we get
logx^a = logy^b
⇒ a(logx) = b(logy)
∴ logx/logy = b/a

৭৬.

3log2 + log5 =?

ক
ক) log20
খ
খ) log10
গ
গ) log40
ঘ
ঘ) log30

ব্যাখ্যা

3log2 + log5
= log2³ + log5
= log8 + log5
= log (8 × 5)
= log40

৭৭.

If log₁₀(2^m + m - 4) = m(1 - log₁₀5), then m =?

ক
ক) 0
খ
খ) 1
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

Question: If log₁₀(2^m + m - 4) = m(1 - log₁₀5), then m =?

Solution:
log₁₀(2^m + m - 4) = m(1 - log₁₀5)
⇒ log₁₀(2^m + m - 4) = m (log₁₀10 - log₁₀5)
⇒ log₁₀(2^m + m - 4) = m log₁₀(10/5)
⇒ log₁₀(2^m + m - 4) = m log₁₀2
⇒ log₁₀(2^m+ m - 4) = log₁₀2^m
⇒ 2^m + m - 4 = 2^m
⇒ m = 4

৭৮.

If 2^{n - 1} + 2^{n + 1} = 640, then n is equal to-

ক
ক) 2
খ
খ) 3
গ
গ) 7
ঘ
ঘ) 8

ব্যাখ্যা

2^{n - 1} + 2^{n + 1} = 640
⇒ 2ⁿ.2^{- 1} + 2ⁿ. 2¹ = 640
⇒ 2ⁿ(1/2) + 2ⁿ .2 = 640
⇒ 2ⁿ(2 + 1/2) = 640
⇒ 2ⁿ (5/2) = 640
⇒ 2ⁿ = (640 × 2)/5
⇒ 2ⁿ = 128 × 2
⇒ 2ⁿ = 256
⇒ 2ⁿ = 2⁸
∴ n = 8

৭৯.

If x, y and z are the sides of a right angled triangle, where ‘z’ is the hypotenuse, then find the value of 1/(log_{x + z}y) + 1/(log_{x - z}y).

ক
1
খ
2
গ
3
ঘ
4
ঙ
None of these

ব্যাখ্যা

Question: If x, y and z are the sides of a right angled triangle, where ‘z’ is the hypotenuse, then find the value of 1/(log_{x + z}y) + 1/(log_{x - z}y).

Solution:
Here x, y and z are the sides of a right angled triangle, so z² = x² + y².

ঋণাত্মক সংখ্যার লগারিদম হয় না বিধায়, সঠিক উত্তর: ঙ) None of these

৮০.

If log_x(144) = 4 then x = ?

ক
2√3
খ
4
গ
3
ঘ
3√2

ব্যাখ্যা

Question: If log_x(144) = 4 then x = ?

Solution:
Given that,
log_x(144) = 4
⇒ x⁴ = 144
⇒ x⁴ = (2√3)⁴
∴ x = 2√3

৮১.

What is the value of: 2³⁰ + 2³⁰ + 2³⁰ + 2³⁰?

ক
16¹²⁰
খ
8³⁰
গ
2³⁰
ঘ
2³²

ব্যাখ্যা

Question: What is the value of: 2³⁰ + 2³⁰ + 2³⁰ + 2³⁰?

Solution:
2³⁰ + 2³⁰ + 2³⁰ + 2³⁰
= 4 × 2³⁰
= 2² × 2³⁰
= 2^{2 + 30}
= 2³²

৮২.

Which of the following are equal in value?
I) 4⁰
II) 1⁴
III) 4¹
IV) 0⁴

ক
ক) III and IV
খ
খ) II and III
গ
গ) I and II
ঘ
ঘ) I and IV

ব্যাখ্যা

I) 4⁰ = 1
II) 1⁴ = 1
III) 4¹ = 4
IV) 0⁴ = 0
So, I and II are equal in value

৮৩.

If log⁡_m243 + log⁡_m81 = 9, find the value of m.

ক
- 2
খ
3
গ
- 4
ঘ
5

ব্যাখ্যা

Question: If log⁡_m243 + log⁡_m81 = 9, find the value of m.

Solution:
Given that,
log⁡_m243 + log⁡_m81 = 9
⇒ log⁡_m(243 × 81) = 9
⇒ log⁡_m19683 = 9
⇒ m⁹ = 19683
⇒ m⁹ = 3⁹
∴ m = 3

৮৪.

If an exponent or index has base 25 and power zero, then which of the following will be its value?

ক
25
খ
5
গ
1
ঘ
250

ব্যাখ্যা

Question: If an exponent or index has base 25 and power zero, then which of the following will be its value?

Solution:
a⁰= 1 (for any non-zero base a)

Now,
= 25⁰= 1

৮৫.

3^x + 3^x + 3^x = ?

ক
9^x
খ
3^2x
গ
3^{x + 1}
ঘ
3x

ব্যাখ্যা

Question: 3^x + 3^x + 3^x = ?

Solution:
3^x + 3^x + 3^x
=3 × 3^x
= 3^{x + 1}

৮৬.

ক
341/13
খ
229/11
গ
336/15
ঘ
None of these

ব্যাখ্যা

Question:

Solution:

৮৭.

If log_n2 = p and log_n5 = q, then log_n250 = ?

ক
p + q²
খ
pq²
গ
p + 2q
ঘ
p + 3q

ব্যাখ্যা

Question: If log_n2 = p and log_n5 = q, then log_n250 = ?

Solution:
Given,
log_n2 = p and log_n5 = q

∴ log_n250 = log_n(2 ×125)
= log_n(2 × 5³)
= log_n2 + log_n5³
= log_n2 + 3 log_n5
= p + 3q

৮৮.

Find the value of n, if 81^{{n - (1/2)}} = 729

ক
0
খ
-1
গ
1
ঘ
2
ঙ
-2

ব্যাখ্যা

Question: Find the value of n, if 81^{{n - (1/2)}} = 729

Solution:
81^{{n - (1/2)}} = 729
⇒ (3⁴)^{{n - (1/2)}} = 3⁶
⇒ 3^{(4n - 2)} = 3⁶
⇒ 4n - 2 = 6
⇒ 4n = 6 + 2
⇒ 4n = 8
∴ n = 2

৮৯.

If 4^{3x + 5} = 1/16^{x + 1}, Find the value of x.

ক
- 2
খ
5/3
গ
4
ঘ
- 7/5

ব্যাখ্যা

Question: If 4^{3x + 5} = 1/16^{x + 1}, Find the value of x.

Solution
4^{3x + 5} = 1/16^{x + 1}
⇒ 2^{2(3x + 5)} = 1/2^{4(x + 1)}⇒ 2^{6x + 10} = 2^{- 4(x + 1)}⇒ 6x + 10 = - 4(x + 1)
⇒ 6x + 10 = - 4x - 4
⇒ 6x + 4x = - 4 - 10
⇒ 10x = - 14
⇒ x = - 14/10
∴ x = - 7/5

৯০.

If log_x(xy) = m, the value of log_y(xy) =?

ক
ক) m
খ
খ) (m - 1)
গ
গ) m/(m + 1)
ঘ
ঘ) m/(m - 1)

ব্যাখ্যা

Question: If log_x(xy) = m, the value of log_y(xy) =?

solution:
log_x(xy) = m
⇒ x^m = xy
⇒ x^m/x = xy/x
⇒ x ^{m - 1} = y
∴ x = y ^{1/m -1}

log_y(xy)
= log_yx + log_yy
= log_yy^{1/(m -1)} + 1
= 1/(m - 1) log_yy + 1
= {1/(m - 1)} + 1
= (1 + m - 1)/(m - 1)
= m/(m - 1)

৯১.

If log₃x+log₉x²+log₂₇x³ =9, then x equals to -

ক
3
খ
9
গ
27
ঘ
81
ঙ
243

ব্যাখ্যা

[ log₍_x^a)(y^b) = b/a log_xy ]

৯২.

49 × 49 × 49 × 49 = 7^?

ক
4
খ
7
গ
8
ঘ
16

ব্যাখ্যা

Question: 49 × 49 × 49 × 49 = 7^?

Solution:
49 × 49 × 49 × 49 = 7^?
⇒ 7²× 7² × 7² × 7² = 7^?
⇒ 7^{2 + 2 + 2 + 2} = 7^?
⇒ 7⁸ = 7^?
? = 8

৯৩.

ক
132
খ
177
গ
185
ঘ
225
ঙ
None of these

ব্যাখ্যা

Question:

Solution:

৯৪.

If (8.97)²× ( 15.05)² ÷ √624.89 = 9ⁿ then the value of n is

ক
ক) 3
খ
খ) 4
গ
গ) 2
ঘ
ঘ) 5

ব্যাখ্যা

(8.97)² × (15.05)² ÷ √624.89 = 9ⁿ
⇒ 9² × 15² ÷ √625 = 9ⁿ
⇒ 81 × 225 ÷ 25 = 9ⁿ
⇒ 81 × 9 = 9ⁿ
⇒ 9³ = 9ⁿ
⇒ n = 3

৯৫.

The value of 5^1/4 × (125)^0.25 is-

ক
√5
খ
5√5
গ
5
ঘ
25
ঙ
None of these

ব্যাখ্যা

Question: The value of 5^1/4 × (125)^0.25 is -

Solution:
5^1/4 × (125)^0.25
= 5^1/4 × (5³)^1/4
= 5^{(1/4 + 3/4)}
= 5^(4/4)
= 5

৯৬.

5^{3x - 2} = 625, find the value of x.

ক
0
খ
3
গ
1
ঘ
2

ব্যাখ্যা

Question: 5^{3x - 2} = 625, find the value of x.

Solution:
5^{3x - 2} = 625
⇒ 5^{3x - 2} = 5⁴
⇒ 3x - 2 = 4
⇒ 3x = 6
∴ x = 2

৯৭.

(0.04)^{- 1.5} = ?

ক
40
খ
125
গ
400
ঘ
250

ব্যাখ্যা

Question: (0.04)^{- 1.5} = ?

Solution:
(0.04)^{- 1.5}
= (4/100)^{- 1.5}
= (1/25)^{- 1.5}
= (1/25)^{- (3/2)}
= (25)^3/2
= (5²)^3/2
= 5^{2 × (3/2)}
= 5³
= 125

৯৮.

Find the value of (3log2 + 2log3)/(log36 + log2)

ক
1
খ
2
গ
4
ঘ
6

ব্যাখ্যা

Question: Find the value of (3log2 + 2log3)/(log36 + log2)

Solution:
3log2 + 2log3
= log2³ + log3²
= log8 + log9
= log(8×9)
= log72

log36+log2
= log(36 × 2)
= log72

(3log2 + 2log3) / (log36 + log2)
=log72 / log72
= 1

৯৯.

Simplify

ক
121
খ
241/38
গ
341/13
ঘ
97/17
ঙ
145/19

ব্যাখ্যা

Question: Simplify
Solution:

১০০.

If log(2a/b) + log(3b/a) = log(a + b), then:

ক
ক) a + b = 1
খ
খ) a + b = 6
গ
গ) a + b = - 1
ঘ
ঘ) a + b = 0

ব্যাখ্যা

log (2a/b) + log(3b/a) = log (a + b)
log{(2a/b) × (3b/a)} = log (a + b)
log6 = log (a + b)
a + b = 6