Logarithms

Concept:Solve logarithmic equation by converting to exponential form and checking domain conditions.Explanation:Given $\log_{x+2}(x^3 - 3x^2 - 6x + 8) = 3$.Convert to exponential: $(x+2)^3 = x^3 - 3x^2 - 6x + 8$.Expand left: $x^3 + 6x^2 + 12x + 8 = x^3 - 3x^2 - 6x + 8$.Cancel $x^3$ and $8$: $6x^2 + 12x = -3x^2 - 6x$.Bring all terms: $9x^2 + 18x = 0 \Rightarrow 9x(x+2) = 0$ so $x = 0$ or $x = -2$.Check domain: base $x+2 > 0$ and $\neq 1$, argument $> 0$.For $x = 0$: base $2$ (valid), argument $8$ (valid). So $x=0$ is a solution.For $x = -2$: base $0$ (invalid). So reject.Only $x = 0$ satisfies, but $0$ is not among the options.Answer:$x = 0$, which corresponds to "None of these".