1. We are given the equation: $$2 \log(x - 5) = -1 + 2 \log(x + 3)$$ 2. The goal is to solve for $x$ exactly. 3. Recall the logarithm property: $$a \log b = \log b^a$$. Using this, rewrite the equation: $$\log (x - 5)^2 = -1 + \log (x + 3)^2$$ 4. Move the constant $-1$ to the left side by rewriting $-1$ as $\log 10^{-1}$ (since $\log 10^{-1} = -1$): $$\log (x - 5)^2 - \log 10 = \log (x + 3)^2$$ 5. Use the logarithm subtraction rule: $$\log a - \log b = \log \frac{a}{b}$$: $$\log \frac{(x - 5)^2}{10} = \log (x + 3)^2$$ 6. Since $\log A = \log B$ implies $A = B$ (for $A,B > 0$), we have: $$\frac{(x - 5)^2}{10} = (x + 3)^2$$ 7. Multiply both sides by 10: $$\cancel{10} \times \frac{(x - 5)^2}{\cancel{10}} = 10 (x + 3)^2$$ $$ (x - 5)^2 = 10 (x + 3)^2$$ 8. Expand both sides: $$(x - 5)^2 = x^2 - 10x + 25$$ $$(x + 3)^2 = x^2 + 6x + 9$$ 9. Substitute expansions: $$x^2 - 10x + 25 = 10(x^2 + 6x + 9)$$ 10. Expand right side: $$x^2 - 10x + 25 = 10x^2 + 60x + 90$$ 11. Bring all terms to one side: $$x^2 - 10x + 25 - 10x^2 - 60x - 90 = 0$$ 12. Simplify: $$-9x^2 - 70x - 65 = 0$$ 13. Multiply both sides by $-1$ to simplify: $$9x^2 + 70x + 65 = 0$$ 14. Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=9$, $b=70$, $c=65$. 15. Calculate discriminant: $$\Delta = 70^2 - 4 \times 9 \times 65 = 4900 - 2340 = 2560$$ 16. Calculate roots: $$x = \frac{-70 \pm \sqrt{2560}}{18} = \frac{-70 \pm 16\sqrt{10}}{18}$$ 17. Simplify fraction: $$x = \frac{-35 \pm 8\sqrt{10}}{9}$$ 18. Check domain restrictions: - $x - 5 > 0 \Rightarrow x > 5$ - $x + 3 > 0 \Rightarrow x > -3$ 19. Only solutions with $x > 5$ are valid. 20. Evaluate approximate values: $$x_1 = \frac{-35 + 8\sqrt{10}}{9} \approx \frac{-35 + 25.3}{9} = \frac{-9.7}{9} = -1.08$$ (invalid) $$x_2 = \frac{-35 - 8\sqrt{10}}{9} \approx \frac{-35 - 25.3}{9} = \frac{-60.3}{9} = -6.7$$ (invalid) 21. No valid solutions satisfy the domain, so the equation has no solution. --- Slug: "log equation" Subject: "algebra" Desmos: {"latex":"y=2 \log(x - 5) - 2 \log(x + 3) + 1","features":{"intercepts":true,"extrema":true}} q_count: 2