(2^x)/2-(2^x)/8=(3^x)/9-(3^x)/27;
2^(x-1)-2^(x-3)=3^(x-2)-3^(x-3);
2^(x-1)(1-1/4)=3^(x-2)(1-1/3);
(3/4)2^(x-1)=(2/3)3^(x-2);
2^(x-1)=(8/9)3^(x-2). Since no power of 2 can be a multiple or power of 3 (and vice versa), the 9 in the denominator of 8/9 must also be the power of 3 given by 3^(x-2), so x-2=2, making x=4. Putting this value in 2^(x-1) gives 8, which conveniently takes care of the numerator of 8/9. So x=4 is the solution.