The use of a Josephson charge qubit as a spectrum analyzer to study the quantum noise of the environment was discussed. The decoherence of the Josephson charge qubit was studied by measuring the energy relaxation and dephasing with the help of single-shot readout. It was also observed that the dominant energy relaxation was a spontaneous emission that was induced by quantum noise coupled to a charge degree of freedom. The results show that the spectral density of the noise was roughly proportional to the qubit excitation energy at high frequencies.