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The age of the universe is about 5 X 10^17 seconds; the shortest light pulse produced in a laboratory (1990) lasted for only 6 X 10^-15second. Identify a physically meaningful time interval approximately halfway between these two on a logarithmic scale.

Physics
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Well, log10(5x10^17) is 17.something and log10(6x10^-15) is -15.something - so you need to look somewhere around log10(x) = approximately.... halfway between those two. +1 seems to be more numerically accurate, so a period of time of the order of 10 seconds but greater, while still less than, 100. After a minute's thought though, I'd just go for 0 and say that a second sort of fits the bill.
5 X 10^17 seconds; the shortest light pulse produced in a laboratory (1990) lasted for only 6 X 10^-15secon… .... Identify a physically meaningful time interval approximately halfway between these two on a logarithmic scale
hmmm. well, the book says 55 seconds. I can't seem to figure out how the authors arrived at that answer. I suppose it is semi-irrelevant. I'll just go with 1.5 minutes. bye bye problem...

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You're just trying to find the halfway point between those two daft numbers. Log of the positive one is 17.69; log of the -ve one is -14.22. Halfway between those is 1.73. 10^1.73 is 54-odd. So do we create a new interval of time or do we opt for a minute or a second? I would go with the second but, hey, what do I know?

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