Log(√(10)) + log(0.01)
explain how to solve without calculator .

We use the rule :    log X^n  =  n * log X
 log 10 = 1  as the base of logarithm is also 10.

log √10  =  log (10)^1/2    =  1/2  log 10  =  1/2 * 1 = 1/2

log 0.01  = log 10^-2  = -2  log 10  = - 2 * 1  = -2 

add the two:   1/2 - 2 = -3/2
 
log 0.01 can be done also as :  log 1/100  =  log 1 - log 100
       = 0 - log 10²  = - 2 * log 10 = - 2  * 1 = - 2

here we used  log ( X / Y )  =  log X - log Y