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.

Question

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.

anonymous · Answer

log(10^2)=2 is 100 seconds, which is about how long I can lift weights. ;D

jamesj · Answer

Ok.

To find the mid-point of the these two times on a log scale, first take their logs...

jamesj · Answer

Hence we have, using log base 10, the log of the two times is
$$ x_1 = 17 + \log 5 , \ \ \ x_2 = \log 6 - 15 $$
Now the mean of these two, the mid point, is just
$$ \frac{x_1 + x_2}{2} = \frac{2 + \log 6 + \log 5}{2} \approx 1  + \log 6 \approx 1.5 $$

jamesj · Answer

So far so good?

anonymous · Answer

yea but book answer is 55 sec

jamesj · Answer

We haven't finished yet.  Now to convert 1.5 back into a time unit we take the reverse operation; i.e., raise 10 to the power of 1.5
$$ 10^{1.5} = 31 $$
So 31 seconds.

jamesj · Answer

Now if you did these calculations precisely, you'd find that the log-mean of the two numbers was closer to 1.739 and 10 raised to that power is 54.8 seconds

jamesj · Answer

Got it?

jamesj · Answer

It's good exercise in algebra to prove that this exactly equivalent to taking the geometric mean of these two numbers.

anonymous · Answer

why did you take a reverse operation when you find the time ?

jamesj · Answer

It's the definition of the "log-mean":
- take the log of both numbers
- find the mean (in the log scale)
- reverse the log; i.e., take the exponent