Chapter 4 - Inverse, Exponential, and Logarithmic Functions - Summary Exercises on Functions: Domains and Defining Equations - Exercises - Page 486: 10

$(-\infty,-7)\cup(3,\infty)$.

The domain requirement for $f(x)=log(\frac{x+7}{x-3})$ is $\frac{x+7}{x-3}\gt0$ which gives $x\gt3$ (numerator and denominator both positive) or $x\lt-7$ (numerator and denominator both negative) , in interval notation $(-\infty,-7)\cup(3,\infty)$.