Question 634999
{{{log(9, (root(3, 3)))}}}
This expression represents "the power of 9 that results in {{{ root(3, 3)}}}". If you're very good with exponents, then you you can figure this our in your head.<br>
If not, then the key to figuring this out is to recognize that both 9 and {{{ root(3, 3)}}} are both powers of 3: {{{9 = 3^2}}} and {{{root(3, 3) = 3^((1/3))}}}. Once we realize this, then we would know that converting the base of the logarithm to 3 will make this problem easier.<br>
The base conversion formula is {{{log(a, (p)) = log(b, (p))/log(b, (a))}}}. We can use this to convert our base 9 log to an expression of base 3 logs:
{{{log(3, (root(3, 3)))/log(3, (9))}}}
The numerator represents "the power of 3 that results in {{{ root(3, 3)}}}. We know that this is 1/3. And the denominator represents "the power of 3 that results in 9". This is known to be 2. Substituting these into our expression we get: 
{{{(1/3)/2}}}
which simplifies (by multiplying the top and bottom of the "big" fraction by 3) to:
{{{ 1/6}}}<br>
P.S. Perhaps now you can look back and see how you could have figured this out "the power of 9 that results in {{{ root(3, 3)}}}" in your head. The power of 9 that results in 3 is 1/2 since 3 is the square root of 9. So we want the exponent for the cube root of the square root of 9:
{{{root(3, sqrt(9)) = root(3, 9^((1/2))) = (9^((1/2)))^((1/3)) = 9^((1/6))}}}