SOLUTION: How do you simplify: 1/2ln9 +( ln6-ln3) and log2 8 square root of x/3y squared

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Question 723513: How do you simplify:
1/2ln9 +( ln6-ln3)
and
log2 8 square root of x/3y squared
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%281%2F2%29ln%289%29+%2B%28+ln%286%29-ln%283%29%29
None of these terms, as they are right now, are like terms so we cannot add and subtract them. (Like logarithmic terms have the same bases and the same arguments. These logs have the same base but the arguments (9, 6 and 3) are all different.)

In addition to adding and subtracting like terms, logarithmic terms can be manipulated by using any of the following properties/rules:
  • log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29
  • log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29
  • log%28a%2C+%28p%5En%29%29+=+n%2Alog%28a%2C+%28p%29%29
  • log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29
The first two properties require that the logs have the same base and coefficients of 1. The third property shows us a way to move the exponent of the argument out in front or a way to move a coefficient into the argument as its argument. (This property is often used to get the coefficient "out of the way" so that the first two properties may be used.) The fourth property is the change of base rule.

In our problem the logs all have the same bases. So we will not be needing the change of base rule to get the bases to match. We can use the third rule on the first log to get the exponent out of the way:
ln%289%5E%281%2F2%29%29+%2B%28+ln%286%29-ln%283%29%29
Since an exponent of 1/2 means square root, we can further simplify the first log:
ln%283%29+%2B%28+ln%286%29-ln%283%29%29
The quick way to the end of the problem is to notice that the first and third logs are now like terms and will cancel each other out. This leaves us with a final answer of:
ln(6)

But you teacher may intend for you to simplify:
ln%283%29+%2B%28+ln%286%29-ln%283%29%29
using the properties instead of just adding the like terms. In the parentheses we can use the second property (since those logs, like ours, have a "-" between them):
ln%283%29+%2B+ln%286%2F3%29
Simplifying the second log:
ln%283%29+%2B+ln%282%29
And finally we can use the first property (because of the "+" between the logs):
ln%283%2A2%29
which simplifies to:
ln(6)

"log2 8 square root of x/3y squared"
I can't help you with this because I have no idea what it is. I can guess that the log's base is 2. But...
  • What part of "8 square root of x/3y squared" is in the argument of the log?
  • What part of "x/3y squared" is inside the square root?
  • Does "8 square root" mean 8 times the square root (as it should) or is it an incorrect way to describe an 8th root (like root%288%2C+x%2F3y%5E2%29)?
Please re-post this question and, at the very least, put parentheses around whatever the argument to the log is and around whatever belongs inside the square (or 8th) root.