1. **State the problem:** Solve for $x$ in the equation $$8^7 = 8^3 \cdot 8^x$$. 2. **Recall the exponent rule:** When multiplying powers with the same base, add the exponents: $$a^m \cdot a^n = a^{m+n}$$. 3. **Apply the rule to the right side:** $$8^3 \cdot 8^x = 8^{3+x}$$. 4. **Rewrite the equation:** $$8^7 = 8^{3+x}$$. 5. **Since the bases are equal and nonzero, set the exponents equal:** $$7 = 3 + x$$. 6. **Solve for $x$:** $$x = 7 - 3 = 4$$. **Final answer:** $$x = 4$$.