✓　　 log₂(log₈x) = log₈(log₂x)　➚　(log₂x)² = __.?

　
log₂(log₈x) = log₈(log₂x)
log₂(log₂x/log₂8) = log₂(log₂x)/log₂8
log₂(log₂x/3) = log₂(log₂x)/3
log₂(log₂x) − log₂3 = 1/3 log₂(log₂x)
log₂(log₂x) − 1/3 log₂(log₂x) = log₂3
2/3 log₂(log₂x) = log₂3
log₂(log₂x) = 3/2 log₂3
log₂(log₂x) = log₂(3^(3/2))
log₂x = 3^(3/2) = √27

(log₂x)² = 27

Since we have to take log of a log, let
x = 8^(8^u) = 2^(3*8^u)
Then
log₂(log₈x) = log₂(8^u) = 3u
log₈(log₂x) = log₈(3*8^u) = log₈3 + u
So
2u = log₈3
Hence
(log₂x)² = [3*8^u]^2
= 9*8^2u
= 9*8^(log₈3)
= 9*3
= 27.

log₂(log₈x) = log₈(log₂x) ………… logₐu = log₂u/log₂a
log₂(log₂x / log₂8) = log₂(log₂x) / log₂8
log₂(log₂x / log₂2³) = log₂(log₂x) / log₂2³ …………log₂2³ = 3 log2
log₂[log₂x / (3 log₂2)] = log₂(log₂x) / (3 log₂2) ………… log₂2 = 1
log₂(log₂x / 3) = log₂(log₂x) / 3 ………… Multiply 3 to the both sides
3 log₂(log₂x / 3) = log₂(log₂x) ……….. 3 log₂(log₂x / 3) = log₂(log₂x / 3)³
log₂(log₂x / 3)³ = log₂(log₂x) ………… If log₂a³ = log₂b, then a³ = b
(log₂x / 3)³ = log₂x …………(log₂x / 3)³ = (log₂x)³ / 3³
(log₂x)³ / 3³ = log₂x ………… 3³ = 27
(log₂x)³ / 27 = log₂x ………… Multiply 27 to the both sides
(log₂x)³ = 27 log₂x ………… Divided by log₂x on the both sides
(log₂x)² = 27