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Just wondering, are there any useful tricks to make estimates of large powers or logarithms just by hand such as for $e^{10}$? Any such ways to get an error less than 1?

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    Well, if you know that the base-10 logarithm of $e$ is roughly $0.434$, then you know that $\log (e^{10}) \approx 4.34$, and as $10^4 = 10000$, we can guess that $e^{10} \approx 20000$. The actual value is about $22026$.2012-11-04

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Many estimation methods exist.
For example, a popular and quick method to estimate the exponential function is using its power series:

$e^x = 1 + x+ \frac{x^2}{2!}+\cdots$ and this series can be used to obtain $\exp$ to a sufficient accuracy.

As for logarithms, using a series, such as the Taylor's series for $\operatorname {artanh}$, combined with pre-computed tables is an optimal way to compute logarithms numerically.