Question

which equation is equivalent to $log_{3}(2x^{4} + 8x^{3}) - 3log_{3}x = 2log_{3}x$?$log_{3}(-x^{3} + 8x^{2}) = log_{3}x^{2}LXB0log_{3}(2x + 8) = log_{3}x^{2}$completewhat are the potential solutions to the equation?$x = -4$ and $x = 2LXB1x = -2$ and $x = 4$$x = 2$ and $x = 4$

Explanation:

Step1: Start with the equivalent equation

We use the confirmed equivalent equation: $\log_3(2x + 8) = \log_3x^2$

Step2: Remove log (same base)

If $\log_b A = \log_b B$, then $A=B$.
$2x + 8 = x^2$

Step3: Rearrange to quadratic form

$x^2 - 2x - 8 = 0$

Step4: Factor the quadratic

$(x - 4)(x + 2) = 0$

Step5: Solve for x

Set each factor to 0:
$x - 4 = 0 \implies x=4$
$x + 2 = 0 \implies x=-2$
(Note: We check domain later, but the question asks for potential solutions, so we include both before verifying validity)

Answer:

$x = -2$ and $x = 4$