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

Question 706863: Log6(1/(4th root(216)))
I need the steps to evaluate the expression down to -3/4
Thanks.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

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:

The argument is the reciprocal of . The exponent for a reciprocal is -1:

Next we check to see if 216 is a power of 6. We know 6 to the first and second powers. Trying we find that it is indeed 216! So we can replace the 216:

In the argument we have a power of a power. The rule for this is to multiply the exponents:

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 . We can actually see the exponent for 6 here.

But if this is not clear, then you can use a property of logarithms, , which allows us to move the exponent of the argument out in front. Using this property on our log we get:

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:

which simplifies to:

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