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?


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

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:

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


$$ t = \sqrt{t_1t_2}$$

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


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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