fourier transform - What's the difference between using DFT, IDFT or DCT to calculate cepstrum of a power spectrum?

I've seen different equations that calculate cepstrum from power spectrum, but the equations are not consistent. Some people use Fourier transform, some use the inverse Fourier transform, and some use the Cosine transform. I.e.:

$$ C(k) = IDFT(\log_{10}P(n)) \\ C(k) = DFT(\log_{10}P(n)) \\ C(k) = DCT(\log_{10}P(n)) \\ $$

For example to calculate LPCC (Linear Prediction Cepstral Coefficients), I see either IDFT and DFT being used equally commonly, while MFCC (Mel Frequency Cepstral Coefficients) uses DCT exclusively.

Are they different? If so what's the different? If not why is there an inconsistency?