Question

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

Answer 1

Since nothing different is shown, we can assume that these logs
are to the base 10.

So the log of a number means:
In order to get that number, what power do you raise  10  to ?

log(√10) = 1/2 . . . . . (10 to the 1/2 power produces √10)

log(0.01) = -2 . . . . . (10 to the -2 power means 1/10² and produces 0.01)

So the sum of those two logs is  (1/2) + (-2) = -(1 and 1/2) .

Answer 2

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
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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