To verify the hypothesis about the common origin of the low-frequency 1/f noise and the quantum f noise recently measured in the Josephson charge qubits, we study the temperature dependence of the 1/f noise and decay of coherent oscillations. The T2 dependence of the 1/f noise is experimentally demonstrated, which supports the hypothesis. We also show that dephasing in the Josephson charge qubits off the electrostatic energy degeneracy point is consistently explained by the same low-frequency 1/f noise that is observed in the transport measurements.