SOLUTION: Write as a single quantity: (1/5)[3log(x+1)+2log(x-1)-log7]

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Question 636518: Write as a single quantity: (1/5)[3log(x+1)+2log(x-1)-log7]
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%281%2F5%29%283log%28%28x%2B1%29%29%2B2log%28%28x-1%29%29-log%28%287%29%29%29%29
There are two ways to combine/condense logarithmic terms:
  • Add or subtract them. Of course adding and subtracting require like terms. Like logarithmic terms have the same bases and the same arguments.
  • Use one of the following properties of logarithms:
    • 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
    These properties require that the bases be the same and the coefficients be 1's.
The three logs in our expression all have the same bases, 10, but they have different arguments. So we will not be able to add or subtract them. The coefficients are not 1's so it seems that we cannot use the properties, either. But fortunately there is a third property of logarithms, q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29, which allows us to "move" the coefficient of a logarithm into the argument as its exponent. So we can use this property to get the coefficients of 1's we need for the other two properties:

Now we can use the first property to combine the first two logarithms. (We use the first one because its logs, like our first two, have a "+" between them:
%281%2F5%29%28log%28%28%28x%2B1%29%5E3%28x-1%29%5E2%29%29-log%28%287%29%29%29%29
Now we can use the second property (because of the "-" between the logs) to combine the remaining two logs:
%281%2F5%29%28log%28%28%28x%2B1%29%5E3%28x-1%29%5E2%2F7%29%29%29
This is a "single quantity" and may be an acceptable answer. But we could use the third property to move the remaining coefficient:
log%28%28%28%28x%2B1%29%5E3%28x-1%29%5E2%2F7%29%5E%28%281%2F5%29%29%29%29
This may be preferred over the first answer. Or we could use the fact that an exponent of 1/5 means 5th root, which in radical form we be:
log%28%28root%285%2C+%28x%2B1%29%5E3%28x-1%29%5E2%2F7%29%29%29%29
Any of the last three expressions should be acceptable. But your teacher may prefer one over the others. My guess would be the last one. It shows the most understanding of how to use the properties and an understanding of fractional exponents.