Since 4 = 22 ---> 4x-1 = (22)x-1 = 22x-2
Since 22x-2 = 22x · 2-2 = 22x · ¼ = ¼ · 22x
4x-1 = 2x + 8 ---> ¼ · 22x = 2x + 8 ---> ¼ · 22x - 2x - 8 = 0
Multiply by 4: ---> 22x - 4 · 2x - 32 = 0
Since 22x = (2x)2 ---> 22x - 4 · 2x - 32 = 0 ---> (2x)2 - 4 · 2x - 32 = 0
Since A2 - 4A - 32 = 0 factors into (A - 8)(A + 4) = 0
Substitute 2x for A:
[ 2x - 8 ] [ 2x + 4 ] = 0
So, either 2x - 8 = 0 or 2x + 4 = 0
---> 2x - 8 = 0 ---> 2x = 8 ---> x = 3 <--- Answer!
---> 2x + 4 = 0 will not give a real answer.
Since 4 = 22 ---> 4x-1 = (22)x-1 = 22x-2
Since 22x-2 = 22x · 2-2 = 22x · ¼ = ¼ · 22x
4x-1 = 2x + 8 ---> ¼ · 22x = 2x + 8 ---> ¼ · 22x - 2x - 8 = 0
Multiply by 4: ---> 22x - 4 · 2x - 32 = 0
Since 22x = (2x)2 ---> 22x - 4 · 2x - 32 = 0 ---> (2x)2 - 4 · 2x - 32 = 0
Since A2 - 4A - 32 = 0 factors into (A - 8)(A + 4) = 0
Substitute 2x for A:
[ 2x - 8 ] [ 2x + 4 ] = 0
So, either 2x - 8 = 0 or 2x + 4 = 0
---> 2x - 8 = 0 ---> 2x = 8 ---> x = 3 <--- Answer!
---> 2x + 4 = 0 will not give a real answer.