SOLUTION: Simplify using logarithmic properties, \log_6 (\frac{216^x}{1296^x})\log_x6

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Question 1207843: Simplify using logarithmic properties,
\log_6 (\frac{216^x}{1296^x})\log_x6
Answer by mccravyedwin(406) About Me  (Show Source):
You can put this solution on YOUR website!
I wish you wouldn't use that notation.  I looked it up and
from what I read it means this.  Let me know if it means
something else:

\log_6 (\frac{216^x}{1296^x})\log_x6

means this, I think:

log%286%2C%28216%5Ex%2F1296%5Ex%29%29%2A%28log%28x%2C%286%29%29%29

Use the rule a%5En%2Fb%5En=%28a%2Fb%29%5En

log%286%2C%28216%2F1296%29%5Ex%29%2A%28log%28x%2C%286%29%29%29

Reduce the fraction, and use the rule log%28a%2C%28b%29%29=1%5E%22%22%2Flog%28b%2C%28a%29%29

log%286%2C%281%2F6%29%5Ex%29%2Aexpr%281%5E%22%22%2F%28log%286%2C%28x%29%29%29%29

Move the exponent x out front:

x%2Alog%286%2C%281%2F6%29%29%2Aexpr%281%5E%22%22%2F%28log%286%2C%28x%29%29%29%29

Write 1%2F6 as 6%5E%28-1%29

x%2Alog%286%2C%286%5E%28-1%29%29%29%2Aexpr%281%5E%22%22%2F%28log%286%2C%28x%29%29%29%29

Move the exponent -1 out front as a negative sign

-x%2Alog%286%2C%286%29%29%2Aexpr%281%5E%22%22%2F%28log%286%2C%28x%29%29%29%29

Use the rule log%28a%2C%28a%29%29=1

-x%2A1%2Aexpr%281%5E%22%22%2F%28log%286%2C%28x%29%29%29%29

-x%5E%22%22%2Flog%286%2C%28x%29%29

Edwin