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Simple Thinking (Posted on 2013-07-09) Difficulty: 1 of 5
Without using a calculator or a computer, determine which of the two numbers 3111 or 1714 is larger.

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The mantissa of the common log of 31 is near .5 as that pair of leading digits is very near the middle of the C or D scale of a slide rule. The characteristic is 1 as it's a 2-digit number. So the log of 31^11 is about 11*1.5 = 16.5.

The mantissa of the log of 2 is about .30. Interpolating from the log of 1, we'd get about (7/10)*.3 = .21, and actually it should be somewhat larger due to logs slope decreasing with higher numbers, but we'll use .21 as conservative. The characteristic for 17 is again 1.  So log(17^14) is conservatively 14*1.21, which is larger than 14*1.2 = 16.8.

So log(17^14) > log(31^11) and therefore 17^14 > 31^11.

  Posted by Charlie on 2013-07-09 12:23:15
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