SOLUTION: How do you simplify this logarithmic function? \log_6 (\frac{216^x}{1296^x})\log_x6

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Question 1206411: How do you simplify this logarithmic function?
\log_6 (\frac{216^x}{1296^x})\log_x6
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
assuming you are given:
log%286%2C+216%5Ex%2F1296%5Ex%29%2Alog%28x%2C6%29

log%286%2C6%5E%28-x%29%29%2Alog%28x%2C6%29

%28log%286%5E%28-x%29%29%2Flog%286%29%29%2A%28log%286%29%2Flog%28x%29%29

%28log%286%5E%28-x%29%29%2Flog%28x%29%29

%28-x%29%2A%28log%286%29%2Flog%28x%29%29


Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.

Starting expression is

    log%286%2C+%28216%5Ex%2F1296%5Ex%29%29 . log%28x%2C6%29.


First factor is 

    log%286%2C+%28216%5Ex%2F1296%5Ex%29%29 = log%286%2C%281%2F6%5Ex%29%29 = -x.


Second factor is 

    log%28x%2C6%29 = 1%2Flog%286%2Cx%29.


The product of two factors is  -x%2Flog%286%2Cx%29.


ANSWER.  The simplified expression is  -x%2Flog%286%2Cx%29.

Solved.