Question 292415
The following properties of logarithms are keys to problems like this:<ol><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></ol>
Let's see how these work on your expression:
{{{log(a, ((x^3*y)/(w^2z^2)))}}}
Start with property #2 (because the argument is a big fraction/quotient):
{{{log(a, (x^3*y)) - log(a, (w^2z^2))}}}
Next, since each argument is a product, we'll use property #1. (Note the use of parentheses. Whenever you replace one term with multiple terms, it is important to sue parentheses.)
{{{(log(a, (x^3)) + log(a, (y))) - (log(a, (w^2)) + log(a, (z^2)))}}}
The first, third and fourth logarithms have exponents on the arguments. So we'll use property #3 on them:
{{{(3log(a, (x)) + log(a, (y))) - (2log(a, (w)) + 2log(a, (z)))}}}
And finally we simplify. (Note how the parentheses help us know that the subtraction in the middle applies to <i>both</i> terms that follow it.)
{{{3log(a, (x)) + log(a, (y)) - 2log(a, (w)) - 2log(a, (z))}}}