SOLUTION: Express as a sum, difference, and product of logarithms, without using exponents logbaseb 4 radical sign x^5b^9/y^7z^10

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Question 269214: Express as a sum, difference, and product of logarithms, without using exponents
logbaseb 4 radical sign x^5b^9/y^7z^10
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28b%2C+%28root%284%2C+%28x%5E5b%5E9%29%2F%28y%5E7z%5E10%29%29%29%29
If this is not your problem, please repost the problem with parentheses around:
  • Radicands (expressions within radicals)
  • Numerators
  • Denominators
  • Exponents

I'm not sure when "product of logarithms" would ever arise but I think I know what the problem is asking. The problem is to rewrite this logarithm as a equivalent expression which a) has no exponents; and b) has arguments which are simply a number or a variable.

To get rid of the exponents we have a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, which allows us to move an exponent in the argument out in front. To further simplify arguments, we have additional 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

These properties, along with a good understanding of exponents, are what we need to do this problem. First, we'll replace the radical with a fractional exponent:
log%28b%2C+%28%28%28x%5E5b%5E9%29%2F%28y%5E7z%5E10%29%29%5E%281%2F4%29%29%29
Now we'll use the first property to move the exponent out in front:
%281%2F4%29%2Alog%28b%2C+%28%28x%5E5b%5E9%29%2F%28y%5E7z%5E10%29%29%29
Next we'll use the third property to rewrite the logarithm of the fraction as a difference of logarithms. (Note how parentheses are used when replacing the single logarithm with the two term difference!)
%281%2F4%29%2A%28log%28b%2C+%28x%5E5b%5E9%29%29+-+log%28b%2C+%28y%5E7z%5E10%29%29%29%29
Next we'll use the second property on each of the two logarithms to split them into an equivalent sum. (Again, note the use of parentheses. These are very important as we will see.)

Next we'll use the first property again to move the exponents in each logarithm out in front:

And finally we will simplify the expression as much as possible. First of all, log%28b%2C+%28b%29%29+=+1 so we get:

or

The minus sign before the parentheses means both terms inside should be subtracted:

And finally we can distribute the 1/4:

and reduce 10/4:

This, I believe, is the form thr problem is looking for. Note how we used properties of logarithms and exponents to "pull apart" the original argument, piece by piece, until the arguments were simple numbers or variables. (And remember to use parentheses! Without them you would not know to distribute the 1/4 and to have a minus in front of the last logarithm.)