The values of $2^{19}$ and $3^{12}$ differ from each other by less than $1.5\%$. This numerical coincidence has an interesting practical consequence: if we begin with any positive real number $a$ corresponding to the frequency of a musical note, and then multiply by $3/2$ ("go up a perfect fifth") twelve times, we end up on (almost) the same note but seven octaves higher. The entire 12-tone (chromatic) musical scale rests on this numerical coincidence. (Another, perhaps less interesting consequence, of this is that we can approximate $\log_2 3 \approx 19/12$, which is accurate to about one part in a thousand.)
Are there other examples of interesting practical consequences of numerical coincidences like this? (Another, simpler coincidence is $5^3 \approx 2^7$, but I'm not sure if that leads to any interesting applications.). By using the phrase "like this" I am being intentionally vague, which is why I am using the soft-question tag; certainly I would be interested in other approximations of the form $a^n \approx b^m$, but I am certainly open to other examples.