We are asked to solve for all values of $x$ in the equation:

$\log_{-2x-3}(3x^2 - 18) = 2$

Step-by-Step Solution:

1. Rewrite the logarithmic equation in exponential form:

   The logarithmic equation $\log_b(A) = C$ can be rewritten as: $A = b^C$ Here, $b = -2x - 3$, $A = 3x^2 - 18$, and $C = 2$. Thus: $3x^2 - 18 = (-2x - 3)^2$

2. Simplify both sides:

   • On the left-hand side: $3x^2 - 18 = 3(x^2 - 6)$

   • On the right-hand side, expand $(-2x - 3)^2$: $(-2x - 3)^2 = (2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9$

   Now the equation becomes: $3(x^2 - 6) = 4x^2 + 12x + 9$

3. Expand and rearrange the equation:

   Expand the left-hand side: $3x^2 - 18 = 4x^2 + 12x + 9$

   Bring all terms to one side: $3x^2 - 18 - 4x^2 - 12x - 9 = 0$ Simplify: $-x^2 - 12x - 27 = 0$ Multiply the whole equation by $-1$ to simplify: $x^2 + 12x + 27 = 0$

4. Solve the quadratic equation:

   Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For $a = 1$, $b = 12$, and $c = 27$: $x = \frac{-12 \pm \sqrt{12^2 - 4(1)(27)}}{2(1)}$ $x = \frac{-12 \pm \sqrt{144 - 108}}{2}$ $x = \frac{-12 \pm \sqrt{36}}{2}$ $x = \frac{-12 \pm 6}{2}$ This gives two solutions: $x = \frac{-12 + 6}{2} = \frac{-6}{2} = -3$ $x = \frac{-12 - 6}{2} = \frac{-18}{2} = -9$

5. Check for restrictions:

   The base of the logarithm, $-2x - 3$, must be positive. This gives the inequality: $-2x - 3 > 0$ Solve for $x$: $-2x > 3$ $x < -\frac{3}{2}$ So, $x$ must be less than $-\frac{3}{2}$. Checking our solutions:

   • $x = -3$ satisfies $x < -\frac{3}{2}$,
   • $x = -9$ also satisfies $x < -\frac{3}{2}$.

Thus, the solutions are $x = -3$ and $x = -9$.

Would you like further details or have any questions?

Related Questions:

1. How is the exponential form derived from a logarithmic equation?
2. What is the quadratic formula and when is it used?
3. How can we verify solutions in logarithmic equations?
4. What is the domain of a logarithmic function with a negative base?
5. How do restrictions on the base of a logarithmic function affect the solution set?

Tip:

Always check for domain restrictions when solving logarithmic equations to ensure all solutions are valid.