ডেইলি কুইজ [২০০ দিন]

5^z + 5^{1 - z} = 6
⇒ 5^z + 5¹/5^z = 6
⇒ 5^z + 5/5^z = 6
⇒ a + 5/a = 6 [ 5^z = a ]
⇒ a² + 5 = 6a
⇒ a² - 6a + 5 = 0
⇒ a² -5a - a + 5 = 0
⇒ a(a - 5) - 1(a - 5) = 0
⇒ (a - 5)(a - 1) = 0
∴ a - 5 = 0
⇒ a = 5
⇒ 5^z = 5¹
⇒ z = 1
অথবা,
a - 1 = 0
⇒ a = 1
⇒ 5^z = 5⁰
⇒ z = 0

5^{4y + 6} = 125^{y + 2} 
⇒ 5^{4y + 6} = 5^{3(y + 2)}
⇒ 5^{4y + 6} = 5^{3y + 6}
⇒ 4y + 6 = 3y + 6
⇒ y = 0

9^q + 9^q + 9^q
= 3 × 9^q
= 3 × (3²)^q
= 3¹ × 3^2q
= 3^{1 + 2q}

(m/n)^{p - 7} = (n/m)^{p - 9} 
⇒ (m/n)^{p - 7} = (m/n)^{9 - p} 
⇒ p - 7 = 9 - p
⇒ 2p = 16
⇒ p = 8

(∛3 × ∛4)¹²
= (∛3 × ∛4)^{3 × 2 × 2} 
= {(∛3 × ∛4)³}^{2 × 2}
= {(3 × 4)²}²
= (9 × 16)²
= 144²
= (12²)²
= 12⁴

∛[∛{(p^m)³}]
= ∛{(p^m)³}^1/3
= ∛(p^m)
= (p^m)^1/3
= p^m/3

243 × 3^{5x - 6} = 9
⇒ 3^{5x - 6} = 9/243
⇒ 3^{5x - 6} = 1/27
⇒ 3^{5x - 6} = (1/3)³
⇒ 3^{5x - 6} = (3)^{- 3}
⇒ 5x - 6 = - 3
⇒ 5x = 3
∴ x = 3/5

(5^{n + 2} + 35 × 5^{n - 1}) / (32 × 5ⁿ)
= (5ⁿ × 5² + 35 × 5ⁿ × 5^{- 1})/(32 × 5ⁿ)
= 5ⁿ(25 + 35/5)/(32 × 5ⁿ)
= (25 + 7)/32
= 1

(256)^3/4 + (243)^2/5 = 146y
⇒ (4⁴)^3/4 + (3⁵)^2/5 = 146y
⇒ {(4⁴)^1/4}³ + {(3⁵)^1/5}² = 146y
⇒ 4³ + 3² = 146y
⇒  64 + 9 = 146y
⇒ 73 = 146y
⇒ y = 73/146
⇒ y = 1/2

a > 0 এবং a ≠ 1 শর্তে a^x = a^y হলে, x = y হবে।

y > 0 এবং y ≠ 1 হলে, log_y1 = 0 (স্বতসিদ্ধ )

log_p(1/32) = - 5
⇒ p^{- 5} = 1/32
⇒ (1/p)⁵ = (1/2)⁵
⇒ 1/p = 1/2
⇒ p = 2

log_z(7/2) = - 3/2
⇒ z^{- 3/2} = 7/2
⇒ 1/z^3/2 = 1/(2/7)
⇒ z^3/2 = 2/7
⇒ (z^3/2)² = (2/7)²
⇒ z³ = 4/49
⇒ z = ∛(4/49)

log_√515625
= log_√55⁶
= log_√5(√5)¹²
= 12log_√5√5
= 12

log_p(ab²/c³)
= log_pa + log_pb² - log_pc³
= log_pa + 2log_pb - 3log_pc
= 3 + 2 × 5 - 3 × 1
= 3 + 10 - 3
= 10

1/32 এর 2 ভিত্তিক লগারিদম
= log₂(1/32)
= log₂(1/2⁵)
=log₂(1/2)⁵
= log₂2^{- 5}
= - 5log₂2
= - 5

log_mn × log_no × log_op × log_pq
= (log_mn × log_no) × log_op × log_pq
= (log_mo × log_op) × log_pq
= log_mp × log_pq
= log_mq