Answer:
x = 3, x = 6
Step-by-step explanation:
Using the rules of logarithms
log x - log y = log ([tex]\frac{x}{y}[/tex] )
log[tex]x^{n}[/tex] ⇔ nlog x
[tex]log_{b}[/tex] x = n ⇔ x = [tex]b^{n}[/tex]
Given
2[tex]log_{3}[/tex] x - [tex]log_{3}[/tex](x - 2) - 2 = 0 ( add 2 to both sides )
[tex]log_{3}[/tex] x² - [tex]log_{3}[/tex](x - 2) = 2
[tex]log_{3}[/tex] ([tex]\frac{x^2}{x-2}[/tex] ) = 2
[tex]\frac{x^2}{x-2}[/tex] = 3² = 9 ( multiply both sides by x - 2 )
x² = 9(x - 2) ← distribute
x² = 9x - 18 ( subtract 9x - 18 from both sides )
x² - 9x + 18 = 0 ← in standard form
(x - 3)(x - 6) = 0 ← in factored form
Equate each factor to zero and solve for x
x - 3 = 0 ⇒ x = 3
x - 6 = 0 ⇒ x = 6
