Log vs. linear frequency scales of Fourier and wavelet transforms I'm trying to understand the difference between the output of a Fourier transform and a wavelet transform. A Fourier transform is done via the following function:

$$\hat{f}(\xi) = \int^\infty_{-\infty}\ f(t)\ e^{-2\pi i t \xi}\ dt$$

Whereas a wavelet transform is going to use this:

$$F(a,b) = \int^\infty_{-\infty}\  f(x)\psi^*_{a,b}(x)\ dx$$

Now, spectrograms use a windowed Fourier transform. When looking at a signal over time, you can get a graph showing you which frequencies were present at any given time. The frequencies are scaled linearly. However, scalograms use a wavelet transform to obtain the same information. I've frequently seen the output scaled logarithmically (in frequency).

Is there something about wavelets that make their output fundamentally logarithmic? I'm having a hard time seeing it in the above formulas. It seems like you set $\xi$ or $a$ &amp; $b$ give you the frequency you want, and you could calculate results for any frequency you desire.
And FFT can be considered a filter bank. The bandwidth of each filter is inversely proportional to the length of the FFT. The width of each filter sets the spacing such that there isn't either too much overlap or gaps between filters.  In an STFT spectrogram, the window width is fixed for the entire FFT, thus the filter spacing is fixed, which turns out to be a linear spacing.

With wavelets the length of each wavelet, being individually adjustable, can be shorter for the higher frequencies, thus have a wider bandwidth, which allows them to be spaced further apart in frequency without leaving gaps between them in the frequency response, thus allowing logarithmic spacing.