SOLUTION: i need help in expanding {{{ log(2,((32) root(3,x))/(y^3)) }}}as much as possible

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Question 374976: i need help in expanding +log%282%2C%28%2832%29+root%283%2Cx%29%29%2F%28y%5E3%29%29+as much as possible
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
+log%282%2C+%28%2832%2Aroot%283%2Cx%29%29%2F%28y%5E3%29%29%29+
For this we will need all three properties of logarithms:
  • log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29
  • log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29
  • log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29

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%282%2C+%2832%2Aroot%283%2Cx%29%29%29+-+log%282%2C+%28y%5E3%29%29+
This argument of the first logarithm is a product so we can use the first property to split the factors into separate logarithms:
+log%282%2C+%2832%29%29+%2B+log%282%2C+%28root%283%2Cx%29%29%29+-+log%282%2C+%28y%5E3%29%29+
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%282%2C+%2832%29%29+%2B+log%282%2C+%28x%5E%281%2F3%29%29%29+-+log%282%2C+%28y%5E3%29%29+
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%282%2C+%2832%29%29+%2B+%281%2F3%29log%282%2C+%28x%29%29+-+3log%282%2C+%28y%29%29+
Last of all, since 32+=+2%5E5 the first logarithm simplifies to 5:
+5+%2B+%281%2F3%29log%282%2C+%28x%29%29+-+3log%282%2C+%28y%29%29+
This, I believe, is the desired answer.