How to make the $ y $-axis of a spectrogram logarithmic?

You can either use the Mel scale:

Spectrogram[audio, Method -> {"MelFrequency", 100, 100, 7000}, 
 ColorFunction -> GrayLevel, AspectRatio -> 1/5]

Or you can construct an Image and apply a non-linear transformation. I've used FourierParameters->{0,1} as it makes the image more crisp and it's a bit easier to notice that they're storm troopers from Star Wars:

spec = SpectrogramArray[audio, FourierParameters -> {0, 1}];
(* cut the data in half and take the Abs *)
img = Image[Map[Abs, Reverse[Transpose[spec][[1 ;; Round[Length[spec]/2]]]], {2}]];
(* move pixels to log position in y-coordinate *)
ImageForwardTransformation[img, {#[[1]], Log[#[[2]]*1308]/Log[1308]/2} &]