2^(x-3) + 2^(x-2) + 2^(x-1) = 448
We know that x^(a-b) = x^a/x^b
==> 2^x/ 2^3 + 2^x/2^2 + 2^x / 2 = 448
==> 2^x / 8 + 2^x/4 + 2^x/2 = 448
Now simplify:
==> (2^x + 2*2^x + 4*2^x)/8 = 448
we know that 448 = 7*2^6
Now multiply by 8:
==> 2^x + 2*2^x + 4*2^x = 7*8*2^6
==> 2^x + 2*2^x + 4*2^x = 7*2^9
Now factor 2^x:
==> 2^x( 1+2+4) = 7*2^9
==> 7*2^x = 7*2^9
divide by 7:
==> 2^x = 2^9
==> x= 9