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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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your question is unclear IMO. How do you estimate?? One way would be to find power series expansion( exists for exp as well as log) and calculating to whatever accuracy you desire... – TheJoker Nov 4 '12 at 17:38
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 $22026$. – ShreevatsaR Nov 4 '12 at 17:47
up vote 4 down vote accepted

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.

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