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The question states

The mean life of an elementary particle pion is $2×10^{-7}$ nanoseconds. The age of the universe is about $4×10{^9}$ years. Identify the time interval that is approximately half way between these two on a logarithmic scale.

I have problem with last part using logarithmic scales. Why can't we simply find an average of the two time intervals? What difference will be made on using logarithmic scales?

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closed as off-topic by WillO, Carl Brannen, Bill N, Yashas, John Rennie Apr 25 at 9:32

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  • $\begingroup$ I'm not sure what your question is. The average of the intervals is a different thing from the interval that is halfway between on a logarithmic scale. So you could find the average of the intervals (it's very close to $2\times10^9$ years), but that is not what the question asks. Does that resolve your confusion? $\endgroup$ – Will Apr 24 at 18:30
  • $\begingroup$ Do you understand the idea of logarithms? If you don't, go ask your teacher. $\endgroup$ – Bill N Apr 24 at 20:45
  • $\begingroup$ I don't know where those numbers come from, but they're wildly incorrect. The neutral pion has a mean lifetime of $8.4\times10^{-17}$ seconds, the charged pions have a much longer mean lifetime of $2.6\times10^{-8}$ seconds. And the age of the universe is $13.8\times10^9$ years. $\endgroup$ – PM 2Ring Apr 24 at 20:50
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You can't just find the average of the 2 times:

$$ \bar t = \frac 1 2 [t_1 + t_2 ]$$

because the question does not ask for an average. It ask for the "half-way point" on a log scale:

$$ \ln t = \frac 1 2 [\ln{t_1} + \ln{t_2}] $$

which is an average of logarithms.

Now if we exponetiate that:

$$ e^{\ln t} = e^{\frac 1 2 [\ln{t_1} + \ln{t_2}]} $$

and simplify:

$$ t = \sqrt{e^{[\ln{t_1} + \ln{t_2}]} } = \sqrt{e^{\ln{t_1}}e^{\ln{t_2}}}$$

or:

$$ t = \sqrt{t_1t_2}$$

There is a name for this, it's called the "geometric average", but your text book doesn't use that term (why not?).

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Start by converting both of these times to seconds. Then consider a log base 10 scale. What power of 10 is half way between the two values?

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