Eulers identity on the other hand...
The OP is just not very interesting. Pi^2 isn’t close enough to 10 to trigger my not-a-coincidence detector.
A much cooler problem of this nature is: why are musical notes generated by powers of the 12th root of 2? I remember seeing a good YouTube video about this on one of those math channels.
So the musical notes thing is by human fiat, which at least pi^2 isn't. Perhaps I'm missing your meaning though?
Some of the character of older music is actually lost because of this. But hey, now all keys work the same.
pi^2 is 9.87. If that counts as close to 10, then half of all numbers are close to some integer. And the significance of 10 is entirely cultural. Nothing in math favours base 10, we use it because the right civilisations were dominant at the right time. Other civilisations counted using base 4, base 5, base 6 or base 12 (e.g. using the five fingers and a closed fist to count up to 6 for each hand).
That one isn't especially difficult. Humans like the combination of fundamental tone and first harmonic (I'm not aware that there's a culture with a musical system that doesn't respect the octave). In the western culture, we found the similarity between those tones so notable that we classify them as harmonically indistinguishable; thus, A3 (220 Hz) is harmonically tied to A4 (440 Hz). Many cultures, our western predecessors included, also appreciated the 2nd harmonic (f3, or, since octaves are now identical, f1.5) as consonant with the fundamental.
Following on to that, they developed a pattern of notes at f(1.5^2), f(1.5^3) etc. Here it can be noted that the first 12 exponents of 1.5 are (renormalized to the [1:2] space as necessary): 1 , 1.5, 1.125, 1.6875, 1.265625, 1.8984375, 1.423828125, 1.06787109375, 1.601806640625, 1.20135498046875, 1.802032470703125, 1.3515243530273438 and 1.0136432647705078.
Note how infuriatingly close that last is to 1; on a violin string (~.328m) that would be a 4mm difference. Human societies have wrestled with that discrepancy as far back as the Pythagoreans. Musicians and composers in western society didn't come to a compromise for the issue until ~late 16th century, when they DEFINED a semitone as the 12th root of two. This adds a little bit of error to every interval besides the octave in favor of consistency across every key.
-1 = e^(i * pi)
ln(-1) = i * pi
ln(-x) = i * pi + ln(x)
log_b(-x) = (i * pi + ln(x)) / ln(b)