Question

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

Answer 1

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

Answer 2

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) .