SOLUTION: If LOGaX = 2 and LOGaY = 3, what does LOGaX^3Y equal?

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Question 239898: If LOGaX = 2 and LOGaY = 3, what does LOGaX^3Y equal?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
If log%28a%2C+%28x%29%29+=+2 and log%28a%2C+%28y%29%29+=+3, what does log%28a%2C+%28x%5E3y%29%29 equal?

Since we know the first two logs, we will find the third one by expressing it in terms of the first two. This is possible because there are properties of logarithms which allow us to manipulate the arguments of logarithms.

First we will use the property log%28r%2C+%28p%2Aq%29%29+=+log%28r%2C+%28p%29%29+%2B+log%28r%2C+%28q%29%29. This allows us to split the log of a product into the sum of the logs of the factors. This will allow us to split apart the x%5E3 and y:
log%28a%2C+%28x%5E3y%29%29+=+log%28a%2C+%28x%5E3%29%29+%2B+log%28a%2C+%28y%29%29
Next we can use the property log%28r%2C+%28p%5Eq%29%29+=+q%2Alog%28r%2C+%28p%29%29. This allows us to move the exponent on the argument of the log to the front of the log as a coefficient:

Now that log%28a%2C+%28x%5E3y%29%29 has been expressed in terms of log%28a%2C+%28x%29%29 and log%28a%2C+%28y%29%29 we can substitute their values: