How do you derive the mantissa of a logarithm?

If your number was say 5280 it is easier to make it into a decimal with one number before the point:

5280 = 5.280 * 10^3

Then use a logarith table to find the logarith of 5280 which, from my table, is 0.7226339.

This is the Mantissa.

Then you look at the exponent on the ten which is 3 (5.280 * 10^3).

This is your characteristic.

Therefore, the common logarithm of 5280 is 3.7226339 where 3 is the characteristic and 0.7226339 is the mantissa.

Hope this helps, message me if you need more info.