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