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

Remove parentheses in numerator.
1(log(1/1000xy^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.log^b(x^n)=nlog^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


log(x)+2log(y)−3