Question 706863
{{{log(6, (1/root(4, 216)))}}}
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.<br>
We'll start by replacing the radical with the appropriate fractional exponent. The exponent for a 4th root is 1/4:
{{{log(6, (1/216^(1/4)))}}}
The argument is the reciprocal of {{{216^(1/4)}}}. The exponent for a reciprocal is -1:
{{{log(6, (216^(-1/4)))}}}
Next we check to see if 216 is a power of 6. We know 6 to the first and second powers. Trying {{{6^3}}} we find that it is indeed 216! So we can replace the 216:
{{{log(6, ((6^3)^(-1/4)))}}}
In the argument we have a power of a power. The rule for this is to multiply the exponents:
{{{log(6, (6^(-3/4)))}}}
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^(-3/4)}}}. We can actually see the exponent for 6 here.<br>
But if this is not clear, then you can use a property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, which allows us to move the exponent of the argument out in front. Using this property on our log we get:
{{{(-3/4)*log(6, (6))}}}
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:
{{{(-3/4)*1}}}
which simplifies to:
{{{-3/4}}}