Question

Log1/3(x) * log1/3(3x-2) = log1/3(3x-2)

Answer 1

Ответ:

1.

Пошаговое объяснение:

$></p> <p><img src=$ или $log_{\frac{1}{3}} (3x -2) = 0$

1)

$\\ \\ log_{\frac{1}{3}} x - 1 = 0\\ log_{\frac{1}{3}} x = 1\\ x = \frac{1}{3}$

2)

$\\ \\ \\ log_{\frac{1}{3}} (3x - 2) = 0\\ 3x - 2 = 1\\ 3x = 3\\ x = 1$

Найдём область допустимых значений:

$\\ \left \{ {{x>0,} \atop {3x-2>0;}} \right. \\ \\ \left \{ {{x>0,} \atop {3x>2;}} \right. \\ \\ \\ \left \{ {{x>0,} \atop {x>\frac{2}{3};}} \right. \\ \\ x > \frac{2}{3}$

$\frac{1}{3}$ не входит в ОДЗ, не является корнем уравнения;

1 - корень уравнения.

Ответ: 1.

Answer 2

To solve the equation log₁/₃(x) * log₁/₃(3x-2) = log₁/₃(3x-2), we can use properties of logarithms.

First, notice that both sides of the equation have a common term, which is log₁/₃(3x-2). We can cancel out this common term from both sides:

log₁/₃(x) * log₁/₃(3x-2) = log₁/₃(3x-2)

Divide both sides by log₁/₃(3x-2):

log₁/₃(x) = 1

Now, we need to express the equation in exponential form to solve for x. Recall that logₐ(b) = c is equivalent to aᶜ = b.

In this case, the base is 1/3, the exponent is 1, and the result is x:

(1/3)¹ = x

x = 1/3

Therefore, the solution to the equation log₁/₃(x) * log₁/₃(3x-2) = log₁/₃(3x-2) is x = 1/3.