Find X

You have:  (x^1/3)⁴  =  16

Then you took the fourth root of both sides:

      this will give you either  x^1/3  =  2    or    x^1/3  =  -2

Now you need to cube both sides:

      (x^1/3)³  =  (2)³          and         (x^1/3)³  =  (-2)³    

              x  =  8                                    x  =  -8

But, unless you are using complex numbers, logs don't have negative numbers for bases, so the only answer is:  x = 8.

You started correctly, just keep going!

\(\displaystyle\log_x16 = \dfrac{4}{3}\\ x^{\frac{4}{3}} = 16\\ x^4 = 16^3\\ \;\;\;\;=4096\\ x= \pm\sqrt[4]{4096}\\ \;\;=\pm 8\)

\(\text{log with base -8 is impossible} \therefore \text{x = 8}\)

Same as geno's answer