#### Answer

$3log_{5}(x) + \frac{1}{2}log_{5}(y) - 3$

#### Work Step by Step

$$log_{5}(\frac{x^{3}\sqrt {y}}{125})$$
Taking care of the exponents seems like a good first step:
$log_{5}(\frac{x^{3}\sqrt {y}}{125}) = log_{5}(\frac{x^{3}y^{\frac{1}{2}}}{5^{3}}) = log_{5}(5^{-3}x^{3}y^{\frac{1}{2}})$
All three terms are being multiplied, so by the Rules of Logarithms, we can expand as follows:
$log_{5}(5^{-3}x^{3}y^{\frac{1}{2}}) = log_{5}(5^{-3}) + log_{5}(x^{3}) + log_{5}(y^{\frac{1}{2}})$
Since each logarithmic term has an associated exponent, we can use the Rules of Logarithms to justify bringing them to the front of the term as constants:
$ log_{5}(5^{-3}) + log_{5}(x^{3}) + log_{5}(y^{\frac{1}{2}}) = -3log_{5}(5) + 3log_{5}(x) + \frac{1}{2}log_{5}(y)$
Finally, when the term of a logarithm is the same as its base, the Rules of Logarithms justify the term equals to 1:
$ -3(log_{5}(5)) + 3log_{5}(x) + \frac{1}{2}log_{5}(y) = -3(1) + 3log_{5}(x) + \frac{1}{2}log_{5}(y)$
Therefore, the final expansion of the original logarithm is:
$$3log_{5}(x) + \frac{1}{2}log_{5}(y) - 3$$