Question 374976
{{{ log(2, ((32*root(3,x))/(y^3))) }}}
For this we will need all three properties of logarithms:<ul><li>{{{log(a, (p*q)) = log(a, (p)) + log(a, (q))}}}</li><li>{{{log(a, (p/q)) = log(a, (p)) - log(a, (q))}}}</li><li>{{{log(a, (p^q)) = q*log(a, (p))}}}</li></ul>
Since the argument of our initial logarithm is a fraction/division we will use the second property to split the numerator and denominator into separate logarithms:
{{{ log(2, (32*root(3,x))) - log(2, (y^3)) }}}
This argument of the first logarithm is a product so we can use the first property to split the factors into separate logarithms:
{{{ log(2, (32)) + log(2, (root(3,x))) - log(2, (y^3)) }}}
Since 1/3 as an exponent means the same thing as a cube root, I am going to replace the cube root with an exponent of 1/3:
{{{ log(2, (32)) + log(2, (x^(1/3))) - log(2, (y^3)) }}}
Now we can use the third property on the two logarithms with exponents on their arguments. (This is why I changed the cube root into a power of 1/3.)
{{{ log(2, (32)) + (1/3)log(2, (x)) - 3log(2, (y)) }}}
Last of all, since {{{32 = 2^5}}} the first logarithm simplifies to 5:
{{{ 5 + (1/3)log(2, (x)) - 3log(2, (y)) }}}
This, I believe, is the desired answer.