Divide numerator and denominator by 8^x to get (1+8^x)/(2^x+4^x)=3/2. Cross multiply: 2[1+(2^x)^3]=3*[2^x+(2^x)^2] Substitute y=2^x and rearrange to 2y^3-3y^2-3y+2=0 By inspection y=-1; Divide by (y+1) to get 2y^2-5y+2=0 which yields y=(5±3)/4=2 or 1/2 x=ln2/ln2=1 is easily verified. x=ln(1/2)/ln2=-1 is also easily verified. x=ln(-1)/ln2=iπ/ln2 is too complex for me to verify.
