Surds, Indices and Logarithm

PrepBank · পাতা ২ / ৫ · ১০১–২০০ / ৪৭১

১০১.

If 5^{(a + 3)} = 25^{(3a - 4)} then the value of a is = ?

ক
11/5
খ
5/2
গ
9/2
ঘ
7/2

ব্যাখ্যা

Question: If 5^{(a + 3)} = 25^{(3a - 4)} then the value of a is = ?

Solution:
5^{(a + 3)} = 25^{(3a - 4)}
⇒ 5^{(a + 3)} = (5²)^{(3a - 4)}
⇒ 5^{(a + 3)} = 5^{(6a - 8)}
⇒ a + 3 = 6a - 8
⇒ 5a = 11
∴ a = 11/5

১০২.

log 36/log6 =?

ক
1
খ
2
গ
3
ঘ
4

ব্যাখ্যা

Question: log 36/log6 =?

solution:
log 36/log6
= log6²/log6
= 2

১০৩.

If log_x625 = 4 then x = ?

ক
ক) 15
খ
খ) 5
গ
গ) 4
ঘ
ঘ) 25

ব্যাখ্যা

log_x625 = 4
Or, x⁴ = 625 = 5⁴
Or, x = 5

১০৪.

Find the value of x if log_x324= 4

ক
ক) 2√3
খ
খ) 3√2
গ
গ) √6
ঘ
ঘ) 6

ব্যাখ্যা

log_x324= 4
⇒ x⁴ = 324
⇒ x⁴ = (3√2)⁴
∴ x = 3√2

১০৫.

If log₇14 + log₇(5x + 1) - 1 = log₇(x + 5), then what is the value of x?

ক
2/3
খ
1/3
গ
3/5
ঘ
3/7

ব্যাখ্যা

Question: If log₇14 + log₇(5x + 1) - 1 = log₇(x + 5), then what is the value of x?

Solution:
log₇14 + log₇(5x + 1) - 1 = log₇(x + 5)
⇒ log₇14 + log₇(5x + 1) - log₇7 = log₇(x + 5)
⇒ log₇[{14 × (5x + 1)}/7] = log₇(x + 5)
⇒ 2(5x + 1) = x + 5
⇒ 10x + 2 = x + 5
⇒ 10x - x = 5 - 2
⇒ 9x = 3
⇒ x = 3/9
∴ x = 1/3

১০৬.

ক
ক) a
খ
খ) 1
গ
গ) a²
ঘ
ঘ) 0

ব্যাখ্যা

Question:

Solution:

১০৭.

The 2^nd and 6^th term of an arithmetic progression are 8 and 20 respectively. What is the 20^th term?

ক
ক) 56
খ
খ) 59
গ
গ) 62
ঘ
ঘ) 65

ব্যাখ্যা

Question: The 2^nd and 6^th term of an arithmetic progression are 8 and 20 respectively. What is the 20^th term?

Solution:
2^nd term, T₂ = a + d = 8....... (1)
6^th term, T₆ = a + (6 - 1)d
Or, a + 5d = 20...... (2)

Subtract (1) from (2) we get,
4d = 12
∴ d = 3

From (1)
a + 3 = 8
∴ a = 5

20^th term = a + (20 - 1)d
= a + 19d
= 5 + (19 × 3)
= 62

১০৮.

Find the value of [2 - 3 × (2 - 3)^{- 1}]^{- 1}.

ক
1/5
খ
- 1/5
গ
5
ঘ
- 5

ব্যাখ্যা

Question: Find the value of [2 - 3 × (2 - 3)^{- 1}]^{- 1}.

Solution:

[2 - 3 × (2 - 3)^{- 1}]^{- 1}

= [2 - 3 × (- 1)^{- 1}]^{- 1}

= [2 - 3 × (- 1)]^{- 1}
= [2 + 3]^{- 1}

= 5^{- 1}

= 1/5

১০৯.

4^{x + 1} = 32, then x = ?

ক
ক) 2
খ
খ) 3
গ
গ) 3/2
ঘ
ঘ) 2/3

ব্যাখ্যা

দেওয়া আছে,
4^{x + 1} = 32
⇒ 2^{2(x + 1)} = 2⁵
⇒ 2(x + 1) = 5
⇒ 2x + 2 = 5
⇒ 2x = 5 - 2
⇒ 2x = 3
∴ x = 3/2

১১০.

What is the value of (1/3)log₁₀125 - 2log₁₀4 + log₁₀32 + log₁₀1

ক
0
খ
1
গ
1/3
ঘ
1/2

ব্যাখ্যা

Question: What is the value of (1/3)log₁₀125 - 2log₁₀4 + log₁₀32 + log₁₀1

Solution:
(1/3)log₁₀125 - 2log₁₀4 + log₁₀32 + log₁₀1
= (1/3)log₁₀5³ - 2log₁₀2² + log₁₀2⁵ + 0
= log₁₀5 - 4log₁₀2 + 5log₁₀2
= log₁₀5 + log₁₀2
= log₁₀10
= 1

১১১.

If (a/b)^{3x - 4} = (b/a)^{x + 2}, then what is the value of x?

ক
1/2
খ
2
গ
3/2
ঘ
4

ব্যাখ্যা

Question: If (a/b)^{3x - 4} = (b/a)^{x + 2}, then what is the value of x?

Solution:
Given, (a/b)^{3x - 4} = (b/a)^{x + 2}⇒ (a/b)^{3x - 4} = (a/b)^{- (x + 2)}⇒ (a/b)^{3x - 4} = (a/b)^{- x - 2}⇒ 3x - 4 = - x - 2
⇒ 3x + x = - 2 + 4
⇒ 4x = 2
∴ x = 1/2

১১২.

If log₁₀5 + log₁₀ (5x + 1) = log₁₀(x + 5) + 1, then what is the value of x?

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 5

ব্যাখ্যা

Question: If log₁₀5 + log₁₀ (5x + 1) = log₁₀(x + 5) + 1, then what is the value of x?

Solution:
log₁₀5 + log₁₀ (5x + 1) = log₁₀(x + 5) + 1
⇒ log₁₀5 + log₁₀(5x + 1) = log₁₀(x + 5) + log₁₀10
⇒ log₁₀{5(5x + 1)} = log₁₀{10(x + 5)}
⇒ 5(5x + 1) = 10(x + 5)
⇒ 25x + 5 = 10x + 50
⇒ 15x = 45
∴ x = 3

১১৩.

log₂√6 + log₂√(2/3) = ?

ক
0
খ
1
গ
2
ঘ
3

ব্যাখ্যা

Question: log₂√6 + log₂√(2/3) = ?

Solution:
log₂√6 + log₂(√2/3)
= log₂[√{6 · (2/3)}]
= log₂√(2 · 2)
= log₂√(2²)
= log₂2
= 1

১১৪.

If log_x(27/64) = - 3, then x = ?

ক
1/3
খ
4/3
গ
3/4
ঘ
2/3

ব্যাখ্যা

Question: If log_x (27/64) = - 3, then x = ?

Solution:
log_x (27/64) = - 3
⇒ x^{- 3} = 27/64
⇒ x^{- 3} = (3/4)³
⇒ x^{- 3} = 1/(4/3)³
⇒ x^{- 3} = (4/3)^{- 3}
⇒ x = 4/3

১১৫.

If log₈x + log₈(1/6) = 1/3, then the value of x is -

ক
ক) 8
খ
খ) 6
গ
গ) 16
ঘ
ঘ) 12

ব্যাখ্যা

Question: If log₈x + log₈(1/6) = 1/3, then the value of x is -

Solution:
Given that
log₈x + log₈(1/6) = 1/3
⇒ log8{x × (1/6)} = 1/3
⇒ log₈(x/6) = 1/3
⇒ 8^1/3 = x/6
⇒ (2³)^1/3= x/6
⇒ 2 = x/6
x = 12

১১৬.

The value of log₅ (125 × 625)/25 is equal to-

ক
5
খ
25
গ
1/5
ঘ
725

ব্যাখ্যা

Question: The value of log₅ (125 × 625)/25 is equal to-

Solution:
log₅ {(125 ⋅ 625)/25}
= log₅ {(5³ ⋅ 5⁴)/5²}
= log₅ 5^{3 + 4 - 2}
= log₅ 5⁵
= 5 log₅ 5
= 5 ⋅ 1
= 5

১১৭.

If 7⁹ + 7⁹ + 7⁹ + 7⁹ + 7⁹ + 7⁹ + 7⁹ = 7^x, what is the value of x?

ক
9
খ
10
গ
12
ঘ
63

ব্যাখ্যা

Question: If 7⁹ + 7⁹ + 7⁹ + 7⁹ + 7⁹ + 7⁹ + 7⁹ = 7^x, what is the value of x?

Solution:
7⁹ + 7⁹ + 7⁹ + 7⁹ + 7⁹ + 7⁹ + 7⁹ = 7^x
⇒ 7.7⁹ = 7^x
⇒ 7¹⁰ = 7^x
∴ x = 10

১১৮.

Find the value of x if log_x 256 = 4.

ক
2
খ
3
গ
5
ঘ
4

ব্যাখ্যা

Question: Find the value of x if log_x 256 = 4.

Solution:
log_x 256 = 4
⇒ x⁴ = 256
⇒ (x²)² = 256
⇒ x² = 16
⇒ x = √16
⇒ x = 4

∴ The value of x is 4.

১১৯.

If 10^a = 1/2, then 10^{- 6a}?

ক
1/64
খ
64
গ
32
ঘ
1/32

ব্যাখ্যা

Question: If 10^a = 1/2, then 10^- ^6a?

Solution:
10^a = 1/2
⇒ (10^a)^-6 = (1/2)^{- 6}
⇒ 10^{- 6a} = 2⁶
∴ 10^{- 6a} = 64

১২০.

If log(p/q) + log(q/p) = log(p + q), then-

ক
ক) p + q = 0
খ
খ) p - q = 1
গ
গ) p = q
ঘ
ঘ) p + q = 1

ব্যাখ্যা

Question: If log(p/q) + log(q/p) = log(p + q), then-

Solution:
log(p/q) + log(q/p) = log(p + q)
⇒ log {(p/q) × (q/p)} = log(p + q)
⇒ log1 = log(p + q)
∴ p + q = 1

১২১.

If x =y^a, y = z^b and z = x^c then what is the value of abc?

ক
1
খ
2
গ
0
ঘ
1/2

ব্যাখ্যা

Question: If x =y^a, y = z^b and z = x^c then what is the value of abc?

Solution:
Given,
x =y^a, y = z^b and z = x^c

Now,
x^c = z
⇒ (y^a)^c = z
⇒ y^ac = z [ (a^m)ⁿ = a^{m × n}]
⇒ (z^b)^ac = z
⇒ z^abc = z¹
⇒ abc = 1

১২২.

ক
12
খ
9/4
গ
15
ঘ
8

ব্যাখ্যা

Question:

Solution:

১২৩.

If m and n are whole numbers such that, mⁿ = 121, then the value of (m - 1)^{n + 1} is-

ক
1
খ
10
গ
121
ঘ
1000
ঙ
None of these

ব্যাখ্যা

Question: If m and n are whole numbers such that, mⁿ = 121, then the value of (m - 1)^{n + 1} is-

Solution:
If mⁿ = 121
∴ mⁿ = 11²

∴ n = 2; m = 11;
then
(m -1) ^{n + 1} = (11 - 1)^{2 +1} = 10³ = 1000

১২৪.

If 9^√x + 9^√x= 162 then, what is the value of x?

ক
5
খ
6
গ
9
ঘ
√5
ঙ
4

ব্যাখ্যা

Question: If 9^√x + 9^√x= 162 then, what is the value of x?

Solution:
Given that,
9^√x + 9^√x = 162
⇒ 9^√x(1 + 1) = 162
⇒ 9^√x × 2 = 162
⇒ 9^√x = 162/2
⇒ 9^√x = 81
⇒ 9^√x = 9²
⇒ √x = 2
⇒ (√x)² = 2²
∴ x = 4

১২৫.

ক
0
খ
1
গ
5
ঘ
60

ব্যাখ্যা

Question:

Solution:

১২৬.

If log₁₀5 = m then log₁₀(1/50) is equal to-

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

ব্যাখ্যা

Given that
log₁₀5 = m

log₁₀(1/50) = log₁₀1 - log₁₀50
= 0 - log₁₀(5 × 10)
= - (log₁₀5 + log₁₀10)
= - (m + 1)

১২৭.

If a^x=b, b^y=c and xyz=1 then what is the value of c^z?

ক
ক) a
খ
খ) b
গ
গ) ab
ঘ
ঘ) a / b

ব্যাখ্যা

Given. xyz=1,a^x=b,b^y=c
Now, b=a^x
=> b^y=a^xy
=> b^yz=a^xyz
=> c^z=a

১২৮.

If x^-7/2= 1/128 then the value of x is:

ক
ক) 1
খ
খ) 2
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

Question: If x^-7/2= 1/128 then the value of x is:

Solution:
x-7/2 = 1/128
1/x^7/2 = 1/2⁷
x^7/2 = 2⁷
(x^7/2)^1/2 = (2⁷)^1/2(x^1/2)^7/2 =2^7/2
x^1/2 = 2
(x^1/2)² = 2²
x = 4

১২৯.

ক
180
খ
294
গ
315
ঘ
322

ব্যাখ্যা

Question:

Solution:

১৩০.

If m and n are natural numbers, then ^m√n is

ক
ক) always irrational
খ
খ) irrational unless n is the mth power of an integer
গ
গ) irrational unless m is the nth power of an integer
ঘ
ঘ) irrational unless m and n are coprime

ব্যাখ্যা

irrational unless n is the mth power of an integer. If m and n are natural numbers, then ^m√n is irrational unless n is mth power of an integer

১৩১.

Find the value of x, if log(x + 5) + log(x - 5) = 4log2 + 2log3

ক
±13
খ
±8
গ
±14
ঘ
±12

ব্যাখ্যা

Question: Find the value of x, if log(x + 5) + log(x - 5) = 4log2 + 2log3

Solution:
Given,
⇒ log(x + 5) + log(x - 5) = 4log2 + 2log3
⇒ log(x + 5)(x - 5) = 4log2 + 2log3 ;[log mn=log m+log n]
⇒ log(x² - 25) = log2⁴+ log3²
⇒ log(x² - 25) = log16 + log9
⇒ log(x² - 25) = log(16 × 9)
⇒ log(x²- 25) = log144
⇒ x²- 25 = 144
⇒ x² = 144 + 25
⇒ x² = 169
⇒ x = ±√169
∴ x = ±13

১৩২.

If log_x(1/9₎ = - 2, then √x = ?

ক
ক) 9
খ
খ) 3√3
গ
গ) √3
ঘ
ঘ) 3

ব্যাখ্যা

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

সমাধান :

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

সুতরাং, √x = √3

১৩৩.

(10^-15 ÷ 10^-4) = ?

ক
ক) 10^{- 19}
খ
খ) 10^{- 11}
গ
গ) 10¹⁹
ঘ
ঘ) 10¹¹
ঙ
ঙ) 10⁶⁰

ব্যাখ্যা

(10^-15 ÷ 10^-4) = ?

Solution:
(10^-15 ÷ 10^-4)
= (10 ^{- 15 + 4})
= 10 ^{- 11}

১৩৪.

If log⁡10 = 1 and log⁡2 = 0.3010, what is the value of log⁡20?

ক
0.6010
খ
0.9030
গ
1.6020
ঘ
1.3010

ব্যাখ্যা

Question: If log⁡10 = 1 and log⁡2 = 0.3010, what is the value of log⁡20?

Solution:
log⁡10 = 1
log⁡2 = 0.3010

∴ log⁡20
= log(2 × 10)
= log2 + log10
= 0.3010 + 1
= 1.3010

১৩৫.

If 2^{n - 1}+ 2^{n + 1} = 320, then the value of n is = ?

ক
7
খ
12
গ
5
ঘ
8

ব্যাখ্যা

Question: If 2^{n - 1}+ 2^{n + 1} = 320, then the value of n is = ?

Solution:
Given that,
2^{n - 1} + 2^{n + 1} = 320
⇒ 2^{n - 1} + 2^{n - 1} . 2² = 320
⇒ 2^{n - 1}(1 + 2²) = 320
⇒ 2^{n - 1} . 5 = 320
⇒ 2^{n - 1} = 320/5 = 64
⇒ 2^{n - 1}= 26
⇒ n - 1 = 6
⇒ n = 6 + 1
∴ n = 7

So the value of n is 7.

১৩৬.

If log_x(16/25) = - 2, then x is equal to-

ক
4/5
খ
1/4
গ
5/4
ঘ
2/3

ব্যাখ্যা

Question: If log_x(16/25) = - 2, then x is equal to-

Solution:
log_x(16/25) = - 2
⇒ x^-2= 16/25
⇒ x^-2 = (25/16)^-1
⇒ x^-2= (5/4)^-2
∴ x = 5/4

১৩৭.

If an exponent or index has base 15 and power zero, then which of the following will be its value?

ক
15
খ
1
গ
0
ঘ
225

ব্যাখ্যা

Question: If an exponent or index has base 15 and power zero, then which of the following will be its value?

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

Now,
= 15⁰= 1

১৩৮.

If log₄[log₃(log₂x)] = 0, then find the value of x.

ক
2
খ
3
গ
5
ঘ
8
ঙ
125

ব্যাখ্যা

Question: If log₄[log₃(log₂x)] = 0, then find the value of x.

Solution:
log₄[log₃(log₂x)] = 0
⇒ log₃(log₂x) = 4⁰ [log_bM = c ⇒ M = b^c]
⇒ log₃(log₂x) = 1
⇒ log₂x = 3¹ [log_bM = c ⇒ M = b^c]
⇒ log₂x = 3
⇒ x = 2³ [log_bM = c ⇒ M = b^c]
∴ x = 8

১৩৯.

What is the value of (log√64) / (3log√4)?

ক
ক) 4
খ
খ) √2
গ
গ) 1
ঘ
ঘ) 0

ব্যাখ্যা

Question: What is the value of (log√64) / (3log√4)?

Solution:
(log√64) / (3log√4)
= log8 / 3log2
= log8 / log2³
= log8 / log8
= 1

১৪০.

যদি log₁₀125 + log₁₀8 = x হয়, তবে x এর মান কত?

ক
ক) 2
খ
খ) 1
গ
গ) 0
ঘ
ঘ) 3

ব্যাখ্যা

প্রশ্ন: যদি log₁₀125 + log₁₀8 = x হয়, তবে x এর মান কত?

সমাধান:
log₁₀125 + log₁₀8 = x
⇒ log₁₀(125 × 8) = x
⇒ log₁₀(1000) = x
⇒ x = log₁₀(1000)
⇒ x = log₁₀(10)³
⇒ x = 3log₁₀10
⇒ x = 3

১৪১.

ক
a + b = 1
খ
a - b = 1
গ
a = b
ঘ
a.a = b.b

ব্যাখ্যা

log(a/b) + log(b/a) = log(a + b)
or, log(a + b) = log{(ab)/(ba)}
or, log(a + b) = log1
or, a + b = 1

১৪২.

ক
8
খ
32
গ
64
ঘ
256

ব্যাখ্যা

Question:

Solution:

১৪৩.

ক
ক) 5
খ
খ) 3
গ
গ) 2
ঘ
ঘ) 1
ঙ
ঙ) 7

ব্যাখ্যা

Hence,
we can write, √(6 + x) = x
⇒ 6 + x = x²
⇒ x² - x - 6 = 0
⇒ x2 - 3x + 2x - 6 = 0
⇒ x(x - 3) + 2(x - 3) = 0
⇒ (x + 2) (x - 3) = 0
⇒ x = -2, 3

Since, we cannot be negative, therefore, x = 3.

১৪৪.

If log₃(x² - 4x) - log₃(x - 4) = 4, Than what is the value of x.

ক
36
খ
405
গ
64
ঘ
81

ব্যাখ্যা

Question: If log₃(x² - 4x) - log₃(x - 4) = 4, Than what is the value of x.

Solution:
Given that,
log₃(x² - 4x) - log₃(x - 4) = 4
⇒ log₃[x(x - 4)/(x - 4)] = 4 ; [log_aM - log_aN = log_a(M/N)]
⇒ log₃x = 4
⇒ x = 3⁴
∴ x = 81

১৪৫.

If X⁵ = 32 and Y³ = 343 then, Y - X = ?

ক
10
খ
14
গ
5
ঘ
7

ব্যাখ্যা

Question: If X⁵ = 32 and Y³ = 343 then, Y - X = ?

Solution:
Given that,
X⁵ = 32
⇒ X⁵ = 2⁵
∴ X = 2
and
Y³ = 343
⇒ Y³ = 7³
∴ Y = 7

Y - X
= 7 - 2
= 5

১৪৬.

Find the value of log₂2 + log₂2² + log₂2³ + ........ + log₂2ⁿ.

ক
n(n + 1)/2
খ
n + 1
গ
n
ঘ
2n
ঙ
None of these

ব্যাখ্যা

Question: Find the value of log₂2 + log₂2² + log₂2³ + ........ + log₂2ⁿ.

Solution:
log₂2 + log₂2² + log₂2³ + ........ + log₂2ⁿ
= log₂2 + 2log₂2 + 3log₂2 + ........ + nlog₂2
= 1 + 2 + 3 + ........ + n
= n(n+1)/2

১৪৭.

Find the value of a,
√(2^a) = 64

ক
- 2
খ
8
গ
12
ঘ
16

ব্যাখ্যা

Question: Find the value of a,
√(2^a) = 64

Solution:
Given,
√(2^a) = 64
⇒ (2^a)^1/2 = 2⁶
⇒ 2^(a/2) = 2⁶
⇒ a/2 = 6
⇒ a = 6 × 2
⇒ a = 12

১৪৮.

25, 36, 49, 81, 121, 169, 225

ক
ক) 36
খ
খ) 49
গ
গ) 121
ঘ
ঘ) 169

ব্যাখ্যা

The numbers are squares of odd natural numbers, starting from 5 up to 15. So, 36 is wrong.

১৪৯.

If x and y are positive integers such that 3^{x + y}= 9⁴ and 2^{x - y} = 16, what is the value of x²- y²?

ক
12
খ
16
গ
24
ঘ
32

ব্যাখ্যা

Question: If x and y are positive integers such that 3^{x + y}= 9⁴ and 2^{x - y} = 16, what is the value of x²- y²?

Solution:
Given that,
3^{x + y} = 9⁴
⇒ 3^{x + y} = (3²)⁴
⇒ 3^{x + y} = 3⁸
∴ x + y = 8 ........(1)

And,
2^{x - y} = 16
⇒ 2^{x - y} = 2⁴
∴ x - y = 4 ....... (2)

Now (1) + (2) than we get,
⇒ (x + y) + (x - y) = 8 + 4
⇒ 2x = 12
⇒ x = 12/2
∴ x = 6

From (1),
6 + y = 8
⇒ y = 8 - 6
∴ y = 2

∴ x² - y² = (6)² - (2)²
= 36 - 4
= 32

১৫০.

If 3^{x + y} = 81 and 3^{x - y} = 9, then what are the values of x and y respectively?

ক
(2, 5)
খ
(3, 1)
গ
(3, 4)
ঘ
(5, 2)

ব্যাখ্যা

Question: If 3^{x + y} = 81 and 3^{x - y} = 9, then what are the values of x and y respectively?

Solution:
Given,
3^{x + y} = 81
⇒ 3^{x + y} = 3⁴⇒ x + y = 4 .......(1)

Again,
3^{x - y} = 9
⇒ 3^{x - y} = 3²⇒ x - y = 2 ........(2)

Now, solving (1) and (2) we get,
x + y + x - y = 4 + 2
⇒ 2x = 6
⇒ x = 3

Now, x + y = 4
⇒ 3 + y = 4
⇒ y = 4 - 3
⇒ y = 1

∴ (x, y) = (3, 1)

১৫১.

If log₁₀(x + 2) + log₁₀(x + 1) = log₁₀(x + 2) + 1, then what is the value of x?

ক
5
খ
- 3
গ
12
ঘ
- 6
ঙ
9

ব্যাখ্যা

Question: If log₁₀(x + 2) + log₁₀(x + 1) = log₁₀(x + 2) + 1, then what is the value of x?

Solution:
Given that,
log₁₀(x + 2) + log₁₀(x + 1) = log₁₀(x + 2) + 1
⇒ log₁₀(x + 2) + log₁₀(x + 1) = log₁₀(x + 2) + log₁₀10 ;[log_aa = 1]
⇒ log₁₀(x + 2) - log₁₀(x + 2) + log₁₀(x + 1) = log₁₀10
⇒ log₁₀(x + 1) = log₁₀10
⇒ x + 1 = 10
∴ x = 9

১৫২.

If √(2ⁿ) = 64, then the value of n is

ক
12
খ
8
গ
6
ঘ
16

ব্যাখ্যা

Question: If √(2ⁿ) = 64, then the value of n is-

Solution:
Given that,
√(2ⁿ) = 64
⇒ (2ⁿ)^1/2 = 2⁶
⇒ 2^n/2 = 2⁶
⇒ n/2 = 6
⇒ n = 2 × 6
∴ n = 12

১৫৩.

ক
25
খ
100
গ
125
ঘ
5

ব্যাখ্যা

Question:

Solution:

১৫৪.

Find the value of log₉59049

ক
9
খ
7
গ
5
ঘ
8
ঙ
6

ব্যাখ্যা

Question: Find the value of log₉59049

Solution:
log₉59049
= log₉9⁵
= 5 × log₉9
= 5 × 1
= 5

১৫৫.

log_3√2(1/18) = ?

ক
- 1
খ
- 2
গ
1
ঘ
0

ব্যাখ্যা

Question: log_3√2(1/18) = ?

Solution:
log_3√2(1/18)
= log_3√2{1/(3√2)}²
= log_3√2 (3√2)^{- 2}
= (- 2)log_3√23√2
= - 2

১৫৬.

ক
2
খ
3
গ
4
ঘ
5

ব্যাখ্যা

Question:

Solution:

১৫৭.

Find the value of x, if 9^{2x + 1} = 81³.

ক
4
খ
2.5
গ
5
ঘ
8

ব্যাখ্যা

Question: Find the value of x, if 9^{2x + 1} = 81³.

Solution:
9^{2x + 1} = 81³
⇒ (3²)^{2x + 1} = (3⁴)³
⇒ 3^{2(2x + 1)} = 3¹²
⇒ 4x + 2 = 12
⇒ 4x = 10
⇒ x = 10/4
∴ x = 2.5

১৫৮.

If a^{2x - 5} = b² and a² = b, what is the value of x?

ক
4
খ
9
গ
9/2
ঘ
18

ব্যাখ্যা

Question: If a^{2x - 5} = b² and a² = b, what is the value of x?

Solution :
Given,
a^{2x - 5} = b² ,
a² = b

so, a^{2x - 5} = (a²)²
⇒ a^{2x - 5} = a⁴
⇒ 2x - 5 = 4
⇒ x = 9/2

১৫৯.

Given,

ক
4/3
খ
4
গ
4^4/3
ঘ
4⁵

ব্যাখ্যা

Question: Given,

Solution:

১৬০.

Find the value of 9⁵ × 9^-3 × 9⁴

ক
ক) 9¹²
খ
খ) 9⁶
গ
গ) 9^{- 60}
ঘ
ঘ) 9⁴

ব্যাখ্যা

Question: Find the value of 9⁵ × 9^-3 × 9⁴

Solution:
9⁵ × 9^-3 × 9⁴
=9^{5 + (- 3) + 4}
=9^{5 - 3 + 4}
=9⁶

১৬১.

If m and n are whole numbers such that mⁿ = 121, then the value of (m - 1)^{n + 1} is = ?

ক
1000
খ
676
গ
121
ঘ
10
ঙ
None of these

ব্যাখ্যা

প্রশ্ন: If m and n are whole numbers such that mⁿ = 121, then the value of (m - 1)^{n + 1} is = ?

সমাধান:
দেওয়া আছে,
mⁿ = 121
⇒ mⁿ = 11²
এখানে, m = 11 এবং n = 2

∴ (m - 1)^{n + 1} = (11 - 1)^{2 + 1}
= 10³
= 1000

১৬২.

log_3√3 27 = x², x =?

ক
√5
খ
√3
গ
√2
ঘ
None

ব্যাখ্যা

Question: log_3√3 27 = x², x =?

Solution:

∴ x = √2

১৬৩.

The average salary of 30 officers in an office is 120 tk and the average salary of laborers is 40 tk. Find the total number of laborers if the average salary of the office is 50 tk.

ক
160
খ
210
গ
180
ঘ
220

ব্যাখ্যা

Question: The average salary of 30 officers in an office is 120 tk and the average salary of laborers is 40 tk. Find the total number of laborers if the average salary of the office is 50 tk.

Solution:
The sum of the salary of officers will be = 30 × 120 = 3600
Let the number of laborers = L

ATQ,
3600 + 40L = 50(30 + L)
⇒ 2100 = 10L
∴ L = 210

১৬৪.

যদি log_x(1/125) = - 3 হয়, তবে x এর মান কত?

ক
2
খ
25
গ
9
ঘ
5
ঙ
100

ব্যাখ্যা

প্রশ্ন: যদি log_x(1/125) = - 3 হয়, তবে x এর মান কত?

সমাধান:
log_x(1/125) = - 3
⇒ x^{- 3} = 1/125 [log_a(b) = c ⇒ a^c = b]
⇒ 1/x³ = 1/125
⇒ x³ = 125
∴ x = 5

১৬৫.

2log₁₀5 + log₁₀8 - (1/2)log₁₀4 = ?

ক
0
খ
1
গ
4
ঘ
2

ব্যাখ্যা

Question: 2log₁₀5 + log₁₀8 - (1/2)log₁₀4 = ?

Solution:

১৬৬.

16^{(2x - 3)} = 32^x, What is the value of x?

ক
2
খ
3
গ
4
ঘ
None of these

ব্যাখ্যা

Question: 16^{(2x - 3)} = 32^x, What is the value of x?

Solution:
Given that,
⇒ 16^{(2x - 3)} = 32^x
⇒ (2⁴)^{(2x - 3)} = (2⁵)^x
⇒ 8x - 12 = 5x
⇒ 8x - 5x = 12
⇒ 3x = 12
⇒ x = 12/3
∴ x = 4

১৬৭.

Solve for x: log₃(x + 5) = 3

ক
13
খ
22
গ
27
ঘ
32

ব্যাখ্যা

Question: Solve for x: log₃(x + 5) = 3

Solution:
Given that,
log₃(x + 5) = 3
⇒ x + 5 = 3³[log_ab = c ⇒ b = a^c]
⇒ x + 5 = 27
⇒ x = 27 - 5
∴ x = 22

১৬৮.

If log_x9/16 = – 1/2 the value of the base is

ক
ক) 16/9
খ
খ) 9/16
গ
গ) 256/81
ঘ
ঘ) 81/256

ব্যাখ্যা

log_x9/16 = – 1/2
⇒ x^-1/2 = 9/16
⇒ √x = 16/9
⇒ x = (16/9)²
∴ x = 256/81

১৬৯.

If 7^b= 343, then the value of 7^{(b - 2)}is:

ক
7
খ
14
গ
21
ঘ
49

ব্যাখ্যা

Question: If 7^b= 343, then the value of 7^{(b - 2)} is:

Solution:
Given that,
7^b = 343
⇒ 7^b = 7³
∴ b = 3

Now,
7^{(b - 2)}
= 7^{(3 - 2)}
= 7¹
= 7

১৭০.

ক
ক) y = 0
খ
খ) x = √y
গ
গ) x = y
ঘ
ঘ) x = y/2

ব্যাখ্যা

প্রশ্ন:

সমাধান:

১৭১.

What is the value of 5 + 4 × 5 + 4 × 5²+ 4 × 5³ + 4 ×5⁴ + 4 × 5⁵?

ক
5⁶
খ
5⁷
গ
5⁸
ঘ
5⁹

ব্যাখ্যা

Question: What is the value of 5 + 4 × 5 + 4 × 5²+ 4 × 5³ + 4 ×5⁴ + 4 × 5⁵?

Solution:
5 + 4 × 5 + 4 × 5²+ 4 × 5³ + 4 ×5⁴ + 4 × 5⁵
= 5 + (5 - 1) × 5 + (5 - 1) × 5²+ (5 - 1) × 5³ + (5 - 1) × 5⁴ + (5 - 1) × 5⁵
= 5 + 5² - 5 + 5³ - 5² + 5⁴ - 5³ + 5⁵ - 5⁴ + 5⁶ - 5⁵
= 5⁶

১৭২.

Find the value of n, if 27^n-(1/3) = 243.

ক
2
খ
3
গ
4
ঘ
5

ব্যাখ্যা

Question: Find the value of n, if 27^{n - (1/3)} = 243.

Solution:
27^{n - (1/3)} = 243
⇒ (3³)^{n - (1/3)} = 3⁵
⇒ 3^{3n - 1} = 3⁵
⇒ 3n - 1 = 5
⇒ 3n = 5 + 1
⇒ 3n = 6
∴ n = 2

১৭৩.

log(a/b) + log(b/c) + log(c/a) = ?

ক
logabc
খ
abc
গ
1
ঘ
0
ঙ
None of these

ব্যাখ্যা

Question: log(a/b) + log(b/c) + log(c/a) = ?

Solution:
log(a/b) + log(b/c) + log(c/a)
= loga - logb + logb - logc + logc - loga
= 0

১৭৪.

log_4√3 48 = x², x³ = ?

ক
2
খ
√2
গ
2√2
ঘ
8

ব্যাখ্যা

Question : log_4√3 48 = x², x³ = ?

Solution :
log_4√3 48 = x²
⇒ log_4√3 (4√3)² = x²
⇒ 2log_4√3 4√3 = x² [ log_a a = 1]
⇒ x² = 2
⇒ x = √2
⇒ x³ = (√2)³
∴ x = 2√2

১৭৫.

Find the logarithm of 1/256 to the base 2√2.

ক
- 16
খ
- 13/5
গ
- 16/3
ঘ
12

ব্যাখ্যা

Question: Find the logarithm of 1/256 to the base 2√2.

Solution:
Let,
log_2√2(1/256) = x
⇒ (2√2)^x = 1/256
⇒ (2√2)^x = 1/2⁸
⇒ (2¹.2^1/2)^x = 2^{- 8}
⇒ (2^3/2)^x = 2^{- 8}
⇒ (2)^3x/2 = 2^{- 8}
⇒ 3x/2 = - 8
⇒ 3x = - 16
∴ x = - 16/3

১৭৬.

log(2x² + 17) = log(x - 3)² হলে, x এর মান কত?

ক
- 4, - 2
খ
- 3, 5
গ
2, - 5
ঘ
5, 2
ঙ
None

ব্যাখ্যা

Question: log(2x² + 17) = log(x - 3)² হলে, x এর মান কত?

Solution:
log(2x² + 17) = log(x - 3)²
⇒ log(2x² + 17) = log(x² - 6x + 9)
⇒ 2x² + 17 = x² - 6x + 9
⇒ x² + 6x + 8 = 0
⇒ x² + 4x + 2x + 8 = 0
⇒ x(x + 4) + 2(x + 4) = 0
⇒ (x + 4)(x + 2) = 0
∴ x = - 4, - 2

১৭৭.

If (m/n)^y-1=(n/m)^{y - 3}, the value of y is -

ক
ক) 1
খ
খ) 2
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

Given that,
(m/n)^{y - 1} = (n/m)^{y - 3}
⇒ (m/n)^{y - 1} = (m/n)^{-(y - 3)}
⇒ y -1 = - (y - 3)
⇒ y -1 =- y +3
⇒ y + y = 3 + 1
⇒ 2y = 4
⇒ y = 4/2
∴ y = 2

১৭৮.

Find the value of .

ক
ক) 2
খ
খ) 3
গ
গ) 1
ঘ
ঘ) 4

ব্যাখ্যা

Question: Find the value of .

Solution:
log₂√6 + log₂√(2/3)
= log₂√(6 × 2/3)
= log₂√2²
= log₂2
= 1

১৭৯.

log₅3 × log₃25 =?

ক
1
খ
2
গ
3
ঘ
5

ব্যাখ্যা

Question: log₅3 × log₃25 =?

Solution:
log₅3 × log₃25
= log₅3 × log₃5²
= log₅3 × 2log₃5
= 2 × log₅3 × log₃5
= 2 × log₅3 × (1/log₅3) [log_ab = 1/log_ba]
= 2

১৮০.

ক
9
খ
12
গ
15
ঘ
18

ব্যাখ্যা

Question:

Solution:

১৮১.

If then find the value of x.

ক
2
খ
√2
গ
2√2
ঘ
1

ব্যাখ্যা

Question: If then find the value of x.

Solution:

১৮২.

ক
0
খ
1/2
গ
1
ঘ
2

ব্যাখ্যা

Question:

Solution:

১৮৩.

If log_x125 = 3 then x = ?

ক
ক) √5
খ
খ) 5
গ
গ) 7
ঘ
ঘ) 9

ব্যাখ্যা

log_x125 = 3
Or, x³ = 125 = 5³
Or, x = 5

১৮৪.

ক
4
খ
8
গ
2ⁿ
ঘ
2^{n + 1}

ব্যাখ্যা

Question:

Solution:

১৮৫.

If 2^a = 3^b and 2^{a + 2} - 3^{b + 1} = √3, then find the value of b.

ক
1
খ
1/2
গ
3
ঘ
√3

ব্যাখ্যা

Question: If 2^a = 3^b and 2^{a + 2} - 3^{b + 1} = √3, then find the value of b.

Solution:
Given,
2^{a + 2} - 3^{b + 1} = √3
⇒ (2^a × 2²) - (3^b × 3¹) = √3
⇒ (3^b × 4) - (3^b × 3) = √3
⇒ 3^b(4 - 3) = √3
⇒ 3^b = 3^1/2
∴ b = 1/2

১৮৬.

What is the value of the following expression?

ক
1
খ
5
গ
60
ঘ
1/60
ঙ
1/2

ব্যাখ্যা

Question: What is the value of the following expression?

Solution:

= log₆₀3 + log₆₀4 + log₆₀5 [∵ 1/log_ba = log_ab]
= log₆₀(3 × 4 × 5) [∵ log_b(m) + log_b(n) = log_b(m × n)]
= log₆₀60
= 1

১৮৭.

ক
5
খ
-5
গ
0
ঘ
3

ব্যাখ্যা

Question:

Solution:

১৮৮.

If x and y are whole numbers such that x^y = 64, then the value of (x - 2)^{y + 1} is = ?

ক
10
খ
216
গ
1000
ঘ
676

ব্যাখ্যা

Question: If x and y are whole numbers such that x^y = 64, then the value of (x - 2)^{y + 1} is = ?

Solution:
দেওয়া আছে,
x^y = 64
⇒ x^y = 8²
এখানে, x = 8 এবং y = 2

এখন,
(x - 2)^{y + 1} = (8 - 2)^{2 + 1} [মান বসিয়ে]
= 6³= 216

১৮৯.

If 2^{2x + 1}= 128 then, what is the value of x?

ক
ক) 7
খ
খ) 6
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

Question: If 2^{2x + 1}= 128 then, what is the value of x?

Solution:
2^{2x + 1} = 128
⇒ 2^{2x + 1} = 2⁷
⇒ 2x + 1 = 7
⇒ 2x = 6
⇒ x = 3

১৯০.

If log_x2 = a and log_x5 = b, then log_x50 =

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

ব্যাখ্যা

Here, log_x50 = log_x(2×25)
= log_x2 + log_x5²
= log_x2 + 2log_x5
= a + 2b [As, log_x2 = a; log_x5 = b]

১৯১.

ক
1
খ
a
গ
a^1/3
ঘ
a³
ঙ
None of these

ব্যাখ্যা

Question:

Solution:

১৯২.

If log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + 1, then what is the value of x ?

ক
1
খ
3
গ
5
ঘ
10
ঙ
15

ব্যাখ্যা

Question: If log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + 1, then what is the value of x ?

Solution:
log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + 1
⇒ log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + log₁₀10
⇒ log₁₀[5(5x + 1)] = log₁₀[10(x + 5)
⇒ 5(5x + 1) = 10(x + 5)
⇒ 5x + 1 = 2x + 10
⇒ 3x = 9
∴ x = 3

১৯৩.

Question:

ক
0
খ
1
গ
a³
ঘ
x
ঙ
None of these

ব্যাখ্যা

Question:

Solution:

১৯৪.

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

ক
2
খ
3
গ
4
ঘ
5

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) = mlog₁₀2
⇒ log₁₀(2^m + m - 4) = log₁₀2^m
⇒ 2^m + m - 4 = 2^m
⇒ m = 4

১৯৫.

If 5√5 × 5³ ÷ 5^-3/2 =5^{a + 2} then what is the value of a?

ক
5
খ
3
গ
4
ঘ
6

ব্যাখ্যা

Question: If 5√5 × 5³ ÷ 5^-3/2 =5^{a + 2} then what is the value of a?

Solution:

১৯৬.

2, 4, 12, 48, 240, (....)

ক
ক) 960
খ
খ) 1440
গ
গ) 1080
ঘ
ঘ) 1920

ব্যাখ্যা

Go on multiplying the given numbers by 2, 3, 4, 5, 6. So, the correct next number is 1440.

১৯৭.

ক
3
খ
4
গ
5
ঘ
3/5

ব্যাখ্যা

Question:

Solution:

১৯৮.

If a^x= b, b^y = c and c^z= a; then the value of xyz is:

ক
ক) -1
খ
খ) 1
গ
গ) 1/abc
ঘ
ঘ) abd

ব্যাখ্যা

c^z = a ⇒ (b^y)^z= a
⇒ (a^x)^yz = a
⇒ a^xyz = a
∴ xyz = 1.

১৯৯.

If log_x(9/16) = −1/2, then x is equal to -

ক
ক) -3/4
খ
খ) 3/4
গ
গ) 81/256
ঘ
ঘ) 256/81

ব্যাখ্যা

Question: If log_x(9/16) = −1/2, then x is equal to -

Solution:
log_x(9/16) = −1/2
⇒ x^-1/2 = 9/16
⇒1/√x = 9/16
⇒ √x = 16/9
⇒ x = (16/9)²
⇒ x = 256/81

২০০.

If log₅y - log₅√y = 2log_y5, then find the value of y.

ক
25
খ
35
গ
10
ঘ
15
ঙ
None of these

ব্যাখ্যা

Question: If log₅y - log₅√y = 2log_y5, then find the value of y.

Solution:
log₅y - log₅√y = 2log_y5