Answer:
x = 3/2
Step-by-step explanation:
4^x - 3^{x-1/2} = 3^{x+1/2} - 2^{2x-1}
4^x + 2^{2x-1} = 3^{x+1/2} + 3^{x-1/2}
2^{2x} + 2^{2x-1} = 3^{x+1/2} + 3^{x-1/2}
2^{2x} + 2^{2x} * 2^{-1} = 3^x+ 3^{1/2} + 3^{x} + 3^{-1/2}
2^{2x}(1 + 2^{-1}) = 3^x(3^{1/2} + 3^{-1/2})
2x ln(2) + ln(1 + 2^{-1}) = x ln(3) + ln(√3 + 1/√3)
2x ln(2) - x ln(3) = ln(√3 + 1/√3) - ln(1 + 1/2)
x(2ln(2) - ln(3)) = ln(√3 + 1/√3) - ln(1 + 1/2)
x = [ ln(√3 + 1/√3) - ln(1 + 1/2) ] / (2ln(2) - ln(3))
x = 3/2.
