2log base of 4 x+ log base of 4 (x-1)=1

First, let's review a couple of basics:

2 x log (something) is the same as log(something²)

The sum of the logs of 2 numbers is the same as the log of their product. 



So    2 log₄(x) = log₄(x²)

From the given equation:        log₄(x²) + log₄(x-1)  =  1

Sum of logs = log of the product:  log₄( [x] [x-1] ) = 1

Raise 4 to the power of each side:         (x) (x-1) = 4

Eliminate parentheses on the left:             x² - x = 4

Subtract  4  from each side:                x² - x - 4 = 0

Mash this through the quadratic formula, and you have

   x = 0.5 + 0.5√17  =   2.562  (rounded)

   x = 0.5 - 0.5√17  =  -1.562  (rounded)