4^x = 256^(y-3)

256 = 4^4

use same base

4^x = (4^4)^(y-3)

= 4^[4(y-3)]

4^x = 4^(4y-12)

equate exponents

x = 4y-12

Your starting point is knowing 4^4 = 256.

4ˣ = 256^(y-3)

  = (4⁴)^(y-3)

  = 4^(4y-12)

take log₄ of both sides

log₄(4ˣ) = log₄(4^(4y-12))

x = 4y-12

4^x = 4^{4y - 12)

x = 4y - 12