Math Master

√(2ⁿ) = 64 
⇒ (2ⁿ)^1/2 = 64 
⇒ 2^n/2 = 2⁶
⇒ n/2 = 6
 n = 12

{(9)^7.5 × (3)^2.5}/(27)^1.5 = 3^x 
{(3²)^7.5 × (3)^2.5}/(3³)^1.5 = 3^x
{3¹⁵ × (3)^2.5}/(3)^4.5 = 3^x
3^x = 3^{15 + 2.5 - 4.5}
3^x =3¹³
x = 13

(x^2/3)/21 = 5/(x^1/3) 
x^2/3.x^1/3 = 5 × 21
x^{(2/3) + (1/3)} = 105
x^{(2 + 1)/3} = 105
x = 105

2^{3x - 9} = 5^{3x - 9}
বা, 2^{3x - 9}/5^{3x - 9} = 1
বা, (2/5)^{3x - 9} = 1
বা, (2/5)^{3x - 9} = (2/5)⁰
বা, 3x - 9 = 0
বা, 3x = 9
বা, x = 3
বা, x/3 = 3/3 
   x/3 = 1

3^{x - 3} = 9a^{x - 5} 
⇒ 3^{x - 3}/9 = a^{x - 5}
⇒ 3^{x - 3}/3² = a^{x - 5}
⇒ 3^{x - 5} = a^{x - 5}
⇒ 3^{x - 5}/a^{x - 5}= 1
⇒ (3/a)^{x - 5} = (3/a)⁰
⇒ x - 5 = 0
∴ x = 5

log₁₀(2m/n) + log₁₀(n/4m) = log₁₀(m + n) 
⇒ log₁₀{(2m/n) × (n/4m)} = log₁₀(m + n) 
⇒ log₁₀(m + n) = log₁₀(1/2)
⇒ m + n = 1/2
 ∴ 2m + 2n = 1

(2²)^{m + 2} = 256
2^{2m + 4} = 2⁸
2m + 4 = 8 
2m = 8 - 4
2m = 4 
m= 2

2¹² + 2¹² + 2¹² + 2¹² 
= 2¹²(1 + 1 + 1 + 1)
= 2¹² .4
= 2¹² .2²
= 2 ^{12 + 2}
= 2¹⁴

log₁₀125 + log₁₀8 = x 
log₁₀(125 × 8) = x
log₁₀1000 = x
10^x = 1000
10^x = 10³
x = 3

36.2^{3x - 8} = 3² 
বা, 36.2^{3x - 8} = 9
বা, 2^{3x - 8} =9/36
বা, 2^{3x - 8} =1/4
বা, 2^{3x - 8} =1/2²
বা, 2^{3x - 8} =2^{- 2}
বা, 3x - 8 = - 2
বা, 3x = 8 - 2 
বা, 3x = 6
     x = 2

log₂(log₃81)
= log₂(log₃3⁴)
= log₂(4log₃3)
=log₂4
= log₂2²
= 2log₂2
= 2 .1 
= 2