Prove that : log4[log2{log2 (log3 81)}] = 0

1.	Prove that : log4[log2{log2 (log3 81)}] = 0
Answer» We know that log_m mⁿ = n log_mm and log_mm = 1 ∴ log_m(m)_n = n x log_mm = n x 1 = n ⇒ log_m(m)ⁿ = n …(i) L.H.S. = log₄[log₂{log₂(log₃ 81)}] = log₄[log₂ {log₂(log₃3⁴)}] (∵ 81 = 3⁴) = log₄[log₂ {(log₂⁴)} [According to equal (i)] = log₄{log₂(log₂2²)} (∵ 4 = 2²) = log₄(log₂2) [According to equal (i)] = log₄(1) (∵ log_mm = 1) = 0 = R.H.S.

Answer»

We know that log_m mⁿ = n log_mm

and log_mm = 1

∴ log_m(m)_n = n x log_mm = n x 1 = n

⇒ log_m(m)ⁿ = n …(i)

L.H.S. = log₄[log₂{log₂(log₃ 81)}]

= log₄[log₂ {log₂(log₃3⁴)}] (∵ 81 = 3⁴)

= log₄[log₂ {(log₂⁴)} [According to equal (i)]

= log₄{log₂(log₂2²)} (∵ 4 = 2²)

= log₄(log₂2) [According to equal (i)]

= log₄(1) (∵ log_mm = 1)

= 0 = R.H.S.

Discussion