SOLUTION: Log6(1/(4th root(216))) I need the steps to evaluate the expression down to -3/4 Thanks.

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Question 706863: Log6(1/(4th root(216)))
I need the steps to evaluate the expression down to -3/4
Thanks.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%286%2C+%281%2Froot%284%2C+216%29%29%29
In general, logarithms are exponents. Base 6 logarithms are exponents for a 6. To find this base 6 log without the help of a calculator, we will need to express the argument as some power of 6.

We'll start by replacing the radical with the appropriate fractional exponent. The exponent for a 4th root is 1/4:
log%286%2C+%281%2F216%5E%281%2F4%29%29%29
The argument is the reciprocal of 216%5E%281%2F4%29. The exponent for a reciprocal is -1:
log%286%2C+%28216%5E%28-1%2F4%29%29%29
Next we check to see if 216 is a power of 6. We know 6 to the first and second powers. Trying 6%5E3 we find that it is indeed 216! So we can replace the 216:
log%286%2C+%28%286%5E3%29%5E%28-1%2F4%29%29%29
In the argument we have a power of a power. The rule for this is to multiply the exponents:
log%286%2C+%286%5E%28-3%2F4%29%29%29
At this point we can (and perhaps should) recognize that the answer is -3/4. The entire expression represents the exponent one would put on a 6 to get 6%5E%28-3%2F4%29. We can actually see the exponent for 6 here.

But if this is not clear, then you can use a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, which allows us to move the exponent of the argument out in front. Using this property on our log we get:
%28-3%2F4%29%2Alog%286%2C+%286%29%29
We should know that for all bases, when the argument is the same as the base the log is equal to 1. So our base 6 log of 6 is a 1:
%28-3%2F4%29%2A1
which simplifies to:
-3%2F4