Will the resulting spectrogram from this algorithm be 1D or 2D?
Spectrograms are two-dimensional, usually calculated from a one dimensional input vector. 

As an example say that you want a spectrogram with k-pts of frequency domain span and M-pts of time-domain span.   With NO overlap an input vector of length N = k x M will fill the spectrogram.  Every k samples a k-pt DFT is performed and makes one of the M time-axis vectors on the spectrogram.

For a running spectrogram with an overlap of k/2, then N/2 input samples will be needed to create a k x M spectrogram, and the time scale will be cut in half.

Usually the logarithm only comes in to scale the amplitude of the output of the DFT.  i.e., each of the k complex samples in the DFT output becomes a scalar value computed from the log of the complex sample magnitude.