Is it not simply a question of saying that 2^10 = 1024 is close to 10^3 - 1000. Take 10th roots of both sides and 10^(3/10) = 1.995262... is close to 2.

And therefore we have 1 = 10^0, 2 ≈ 10^0.3, 4 ≈ 10^0.6, 8 ≈ 10^0.9, 16 ≈ 10^1.2, 32 ≈ 10^1.5, 64 ≈ 10^1.8, 128 ≈ 10^2.1, 256 ≈ 10^2.4, and 512 ≈ 10^2.7.

Divide by 10s and we get 1 = 10^0, 2 ≈ 10^0.3, 4 ≈ 10^0.6, 8 ≈ 10^0.9, 1.6 ≈ 10^0.2, 3.2 ≈ 10^0.5, 6.4 ≈ 10^0.8, 1.28 ≈ 10^0.1, 2.56 ≈ 10^0.4, and 5.12 ≈ 10^0.7.

Reorder and we've got the initially surprising 1 = 10^0, 1.28 ≈ 10^0.1, 1.6 ≈ 10^0.2, 2 ≈ 10^0.3, 2.56 ≈ 10^0.4, 3.2 ≈ 10^0.5, 4 ≈ 10^0.6, 5.12 ≈ 10^0.7, 6.4 ≈ 10^0.8 and 8 ≈ 10^0.9.

Then work the 5^10 and 7^13 examples to show that it happens whenever a^b is close to a power of 10.

Leave out the other base observation.