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Eleven Square Roots in a Logarithm (Posted on 2006-09-05) Difficulty: 2 of 5
Presto the Mathematical Magician says, quite correctly, that ln(x), the natural logarithm (to the base e=2.718...) of x, is magically well-approximated by 2047(x1/2048 - 1). Hence logarithms can be calculated with fair accuracy using a primitive calculator that only does square roots along with basic arithmetic.

What is behind Presto's magic?

By the same token, log(x), the common (base 10) logarithm of x, may be approximated by the similar formula K(x1/2048 - 1) for a suitable value of K. For values of x between 1 and 10, explore the accuracy of this approximation, and that of similar formulas of the type K(x1/N-1) where N=2n, under the assumption that a 10-digit calculator is being used to compute the repeated square roots. What values for K and n would you recommend when a 10-digit calculator is being used?

See The Solution Submitted by Richard    
Rating: 3.3333 (3 votes)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(2): Not quite a spoilerJLo2006-09-24 15:25:53
Solutionre: Not quite a spoilervswitchs2006-09-24 04:49:53
Some ThoughtsFew other thoughtsJLo2006-09-10 13:51:06
some thoughtsArt M2006-09-06 17:30:49
For the second part.Charlie2006-09-05 15:55:38
Not quite a spoilervswitchs2006-09-05 14:52:28
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