Let  `log_2 16` = n.

Then , `2^n = 16 = 2^4`

`rArr    n = 4`

`:.`      `log_2 16 = n`

log(0.57) =  `overline(1)`.756    `rArr  log 57 =  1.756  `   [ `because`  mantissa will remain the same ]

`:.`     log 57 +` log (0.57)^3 + log sqrt(0.57)`

=`log57 + 3 log (57/100) + log (57/100)^(1//2)`

=`log 57 + 3 log 57 - 3 log 100 + 1/2 log 57 -  1/2 log 100`

=`9/2 log 57 - 7/2 log 100 `

=`9/2 xx 1.756 - 7/2 xx 2` =  7.902 - 7 =  0.902.

`log5^20` = 20 log 5

=`20 xx [log (10/2)]` =  20(log 10 - log 2)

= 20(1 -  0.3010) = 20 x 0.6990 = 13.9800.

`:.`     Characteristic = 13 , Hence , the number of digit in `5^20` is 14.

`log4^50` = 50log 4 = 50log `2^2` = (50 x 2) log 2 = 100 x log 2 = (100 x 0.30103) =  30.103.

`:.`       Characteristic = 30 , Hence , the number of digit  in `4^50` = 31.

`log (2^64)` = 64 x log 2 = (64 x 0.30103) =  19.26592

Its characteristic is 19 . Hence , the number of digit in `2^64` is  20.

`log_10 (2.8)` =  `log_10 (28/10)` =  `log_10 28 - log_10 10`

=`log_10 (7 xx 2^2) - 1`

=`log_10 7 + 2log_10 2 - 1`

= 0.8451 + 2 x 0.3010 - 1 = 0.8451 + 0.602 - 1 =  0.4471.

`log_10 (1.5)`

=`log_10 (3/2)`

=`log_10 3 - log_10 2`

=(0.4771 - 0.3010) = 0.1761

`log_5 512` = `(log 512)/(log 5)` = `(log 2^9)/(log(10/2))` = `(9 log 2)/(log 10 - log 2)`

=`((9 xx 0.3010))/(1 - 0.3010)`

= `2709/0.699`

=`2709/699` = 3.876.

`log_10 25` = log`_10(100/4)`

=`log_10 100 - log_10 4`

=`2 -  2log_10 2`

= (2 - 2 x  0.3010)   = (2 - 0.6020) = 1.3980