Solution

In this example, we need to unpack some radicals, so I will state here a general reminder that: \[\sqrt[n]{A}=A^{\frac{1}{n}}\] So any radical can be transformed into an exponent. This is a highly useful property when combined with logarithms: \[ \solve{ \log_7\left(\sqrt[3]{x^2-16}\right)&=& \log_7\left((x^2-16)^{\frac{1}{3}}\right)\\ &=&\frac{1}{3}\log_7(x^2-16)\\ &=&\frac{1}{3}\log_7((x+4)(x-4))\\ &=&\frac{1}{3}(\log_7(x+4)+\log_7(x-4))\\ &=&\frac{1}{3}\log_7(x+4)+\frac{1}{3}\log_7(x-4) } \] Don't forget to attempt to factor within the logarithms in order to use the Sum or Difference properties of logarithms in order to expand the expression as much as possible.