Answer (my view): If it's an inverse f(-1)(x) = 6^x/2
Explanation: The function is f(x) = log6(2x), an inverse of a function (i. e: f(-1)(x)) can be defined as a function that takes an output of it's main function (i. e: f(x)) as its input and returns the original input plotted in the main function. Any logarithm can be interpreted as an exponentiation, so log6(2x) would be equal to 6^f(x) = 2x, the function f(-1)(x) would take this exponent of 6 returned by the main function and return a half of the base of the function times 1/2, letting this exponent (output of the main function) be x, f(-1)(x) = 6^x/2.
Algebraically, however: f(x) = log6(2x), and f(f^(-1)(x)) = x.
So log6(2(f^(-1)(x))) = x, then 6^x = 2(f^(-1)(x)), (6^x)/2 = (f^(-1)(x)).