Expand the following logarithm: log of the square root of xy divided by 1000

Remove parentheses in numerator.

1 (log (1/1000 x y^2))

The logarithm of a product is equal to the sum of the logarithms of each factor (e. g. log (xy) = log (x) + log (y)). The logarithm of a division is equal to the difference of the logarithms of each factor (e. g. log (x/y) = log (x) - log (y)).

1 (log (x) + log (y^2) - log (1000))

The exponent of a factor inside a logarithm can be expanded to the front of the expression using the third law of logarithms. The third law of logarithms states that the logarithm of a power of x is equal to the exponent of that power times the logarithm of x (e. g. lo g^b (x^n) = nlo g^b (x)).

log (x) + 1 ((2log (y))) - log (1000)

Remove the extra parentheses from the expression 1 ((2log (y))).

log (x) + 2log (y) - log (1000)

The logarithm base 10 of 1000 is 3.

log (x) + 2log (y) - ((3))

Simplify.

log (x) + 2log (y) - 3

Answer:

log (x) + 2log (y) - 3