4^x-3^ (x-1/2) = 3^ (x+1/2) - 2^ (2x-1)

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.