1. **State the problem:** Solve the logarithmic equation $$\log_4(2x) = \log_4(63 - x^2)$$. 2. **Recall the logarithm property:** If $$\log_a(M) = \log_a(N)$$ and the base $$a > 0, a \neq 1$$, then $$M = N$$, provided both $$M$$ and $$N$$ are positive. 3. **Apply the property:** Set the arguments equal: $$2x = 63 - x^2$$ 4. **Rearrange the equation:** $$x^2 + 2x - 63 = 0$$ 5. **Factor the quadratic:** $$x^2 + 2x - 63 = (x + 9)(x - 7) = 0$$ 6. **Solve for $$x$$:** $$x + 9 = 0 \Rightarrow x = -9$$ $$x - 7 = 0 \Rightarrow x = 7$$ 7. **Check domain restrictions:** - Argument of $$\log_4(2x)$$ requires $$2x > 0 \Rightarrow x > 0$$ - Argument of $$\log_4(63 - x^2)$$ requires $$63 - x^2 > 0 \Rightarrow x^2 < 63 \Rightarrow -\sqrt{63} < x < \sqrt{63}$$ 8. **Check each solution:** - For $$x = -9$$: $$2(-9) = -18 < 0$$, invalid. - For $$x = 7$$: $$2(7) = 14 > 0$$ and $$63 - 7^2 = 63 - 49 = 14 > 0$$, valid. 9. **Final solution set:** $$\{7\}$$.