To solve this logarithmic equation, we can use the properties of logarithms to condense the equation.

First, we can rewrite the equation using the property log (a) - log(b) = log(a/b):

log4(x) - log16(x) = 1/4
log4(x/16) = 1/4

Now, we can rewrite log4 as log base 2 (since 4 = 2^2):

log base 2 (x/16) = 1/4

Next, we can rewrite the equation using the definition of logarithms:

2^(1/4) = x/16

Now, we can solve for x:

2^(1/4) = x/16
2^(1/4) = (2^4)/16
2^(1/4) = 2^3
1/4 = 3

Therefore, there is no solution to the equation log4(x) - log16(x) = 1/4.