Note: `log_32 x = 0.8 ⇔ x = (32)^0.8` `= (2^5)^4/5` `= 2^4` `= 16.`

Note: `log_4 x + log_2 x` `= 6 ⇔ (log x)/(log 4) + (log x)/(log 2)` `= 6` `:. (log x)/(2 log 2) + (log x)/(log 2)` `= 6 ⇔ 3 log x` `= 12 log 2` `or log x` `= 4 log 2 ⇔ log x` `= log ^24 or x = 2^4` `= 16.`

Note: `log x + log_8 1/6` `= 1/3 ⇔ (1og x)/(log 8) + (log 1/6)/(log 8)` `= 1/3` `:. log x + log 1/6` `= 1/3 log 8` `or log x + log 1/6` `= log (8^1/3)` `= log (2^3)^1/3.` `:. log x + log 1/6` `= log 2` `or log x` `= log 2 - log 1/6` `= log (2 xx 6/1)` `= log 12.` `:. x = 12.`

Note: `log ^4^50` `= 50 log 4` `= 50 log 2^2` `= (50 xx 2) log 2` `= 100 xx log 2` `= (100 xx 0.30103)` `= 30.103.` :. Characteristic`= 30.` Hence the number of digits in`4^50` is `31`

Note: `log_5^20` `= 20 log 5` `= 20 xx [log (10/2)]` `= 20 xx [log 10 - log 2]` `= 20 xx [1 - 0.3010]` `= 20 xx .6990` `= 13.9800` :. Characteristic`= 13`. :. Number of digits in`5^20` is `14`.

Note: Let log`(- 1/3)^81` `= x.` Then,` (- 1/3)^x` `= 81 = 3^4` `= (- 3)^4` `= (- 1/3).` `:. x = - 4.`

Note: Let` log2√3 (1728) = x.` Then, `(2√3)^x = 1728` `= (12)^3 = [(2√3)^2]^3` `= (2√3)^6.` `:. x = 6`

Note: Let `log_5 625 = x.` Then,`5^x = 625` `= 5^4 or x = 4.` Let `log_2 (log_5 625) = y.` Then,`log_2 (4) = y` `or 2^y = 4 = 2^2.` So,`y = 2.` :.`log_2 (log_5 625) = 2.`

Note: `1/3 log_10 125 - 2log_10^4 + log_10^32` `= log_10 (125)^1/3 - log_10 (4^2) + log_10 ^32` `= log_10 16 + log_10^32` `= log_10 ((5 xx 32)/(16))` `= log_10^10` `= 1.`

Note: Not available