Chapter 4 - Section 4.5 - Exponential and Logarithmic Equations - 4.5 Exercises - Page 447: 44

$x=-4$

Changing to exponential form, the given expression, $ \log_7(x^3+65)=0 $ is equivalent to \begin{array}{l}\require{cancel} x^3+65=7^0 \\\\ x^3+65=1 \\\\ x^3=1-65 \\\\ x^3=-64 \\\\ x=\sqrt[3]{-64} \\\\ x=\sqrt[3]{(-4)^3} \\\\ x=-4 .\end{array}