SOLUTION: write 3log u - log v - 4log z expression as a single logarithm

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Question 206233: write 3log u - log v - 4log z expression as a single logarithm
Found 2 solutions by stanbon, Theo:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
write 3log u - log v - 4log z expression as a single logarithm
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log u^3 -log v - log z^4
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= log[u^3/(vz^4)]
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Cheers,
Stan H.

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your problem is stated as 3%2Alog%28%28u%29%29+-+log%28%28v%29%29+-+4%2Alog%28%28z%29%29
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There are 3 basic rules of logarithms.
you will be using rules 2 and 3 as shown below.
rule number 1 does not apply to this problem
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rule number 1 states that log%28%28a%2Ab%29%29+=+log%28%28a%29%29+%2B+log%28%28b%29%29
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rule number 2 states that log%28%28a%2Fb%29%29+=+log%28%28a%29%29+-+log%28%28b%29%29
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rule number 3 states that log%28%28a%5En%29%29+=+n%2Alog%28%28a%29%29
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your equation of 3%2Alog%28%28u%29%29+-+log%28%28v%29%29+-+4%2Alog%28%28z%29%29 can be first converted to log%28%28u%5E3%29%29+-+log%28%28v%29%29+-+log%28%28z%5E4%29%29 using rule number 3 in reverse.
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your equation of log%28%28u%5E3%29%29+-+log%28%28v%29%29+-+log%28%28z%5E4%29%29 can be first converted to log%28%28u%5E3%2Fv%29%29+-+log%28%28z%5E4%29%29 and then converted to log%28%28%28%28u%5E3%2Fv%29%29%2F%28z%5E4%29%29%29 using rule number 2 in reverse if we apply the rule successively as we scan from left to right which we can do as long as there are no parentheses telling us to do otherwise. Anything within parentheses is always done first. That is not the case here since we have no parentheses in your original problem statement.
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your final form is log%28%28%28%28u%5E3%2Fv%29%29%2F%28z%5E4%29%29%29
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to prove that this is true, we'll untangle it using the rules 1 and 2 of logarithms and we should be able to get back to the original equation.
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your answer of log%28%28%28%28u%5E3%2Fv%29%29%2F%28z%5E4%29%29%29 becomes log%28%28u%5E3%2Fv%29+-+log%28%28z%5E4%29%29%29 using rule number 2
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this becomes log%28%28u%5E3%2Fv%29+-+4%2Alog%28%28z%29%29%29 using rule number 3.
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this becomes %28log%28%28u%5E3%29%29+-+log%28%28v%29%29%29+-+4%2Alog%28%28z%29%29%29 using rule number 2.
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this becomes %283%2Alog%28%28u%29%29+-+log%28%28v%29%29%29+-+4%2Alog%28%28z%29%29%29 using rule number 3.
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your original problem stated as 3%2Alog%28%28u%29%29+-+log%28%28v%29%29+-+4%2Alog%28%28z%29%29%29
is equivalent to my final version of your original problem as %283%2Alog%28%28u%29%29+-+log%28%28v%29%29%29+-+4%2Alog%28%28z%29%29%29 because my parentheses, in this case, do not alter the order of the operations to be performed.
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