Question

1. solve $log_{3}(5x + 2) = log_{3}(3x - 6)$

Explanation:

Step1: Set arguments equal (log property)

If $\log_b(a) = \log_b(c)$, then $a=c$.
$5x + 2 = 3x - 6$

Step2: Isolate x terms

Subtract $3x$ from both sides.
$5x - 3x + 2 = -6$
$2x + 2 = -6$

Step3: Isolate constant terms

Subtract 2 from both sides.
$2x = -6 - 2$
$2x = -8$

Step4: Solve for x

Divide by 2 on both sides.
$x = \frac{-8}{2} = -4$

Step5: Check domain validity

Logarithm arguments must be positive:
For $5x+2$: $5(-4)+2=-18$, which is negative.
For $3x-6$: $3(-4)-6=-18$, which is negative.
Negative values are not allowed in logarithms, so this solution is extraneous.

Answer:

No real solution