exponents

2^(2^x) = 16 ^(16^16)

Take the log of both sides

log 2^(2^x)  = log(16)^(16^16)   and we can write

(2^x) log 2  =  (16^16)log 16

And we can write

2^x / (16^16)  = log 16/ log 2

2^x / 16^16  =  log 2^4  / log 2

2^x / 16^16  =  4 log 2 / log 2

2^x /  16^16  =  4    rearrange  as

2^x / 4  =  16^16

2^x / 2^2  = 16^16

2^ (x - 2) = (2^4)^16

2^(x - 2) = 2^(64)      we can solve for the exponents

x - 2   = 64

x   =  66