Question

2lg(2-x)^2 -lg(x-2) *lg(x-2)=4

Answer 1

ОДЗ: $(2-x)^2 >0; x-2>0$
$2-x \neq 0; x>2$
$x>2$


$2lg(2-x)^2-lg(x-2)*lg(x-2)=4$
$A^2=(-A)^2$
$2lg(x-2)^2-lg^2 (x-2)=4$
$log_a b^n=n*log_a b$
$2*2lg(x-2)-lg^2 (x-2)=4$
$4lg(x-2)-lg^2(x-2)=4$
делаем замену
$lg(x-2)=y$
$4y-y^2=4$
$y^2-4y+4=0$
$(y-2)^2=0$
$y-2=0$
$y=2$
возвращаемся к замене
$lg(x-2)=2$
$x-2=10^2=100$
$x=100+2=102$
ответ: 102 

Answer 2

To solve the equation 2lg(2-x)^2 - lg(x-2) * lg(x-2) = 4, we can follow these steps:

Step 1: Simplify the expression using logarithmic properties.

Recall the following logarithmic properties:

1. lg(a) + lg(b) = lg(ab)
2. lg(a) - lg(b) = lg(a/b)
3. (lg(a))^n = lg(a^n)

Applying these properties, we can simplify the equation:

2lg(2-x)^2 - lg(x-2) * lg(x-2) = 4 lg((2-x)^2)^2 - lg((x-2)^2) = 4 lg((2-x)^4) - lg((x-2)^2) = 4 lg((2-x)^4 / (x-2)^2) = 4

Step 2: Convert the logarithmic equation to an exponential equation.

Using the definition of logarithms, we can rewrite the equation in exponential form:

(2-x)^4 / (x-2)^2 = 10^4 (2-x)^4 / (x-2)^2 = 10000

Step 3: Solve the exponential equation.

To solve the equation, let's multiply both sides by (x-2)^2 to eliminate the denominator:

(2-x)^4 = 10000 * (x-2)^2

Expand the left side of the equation:

16 - 32x + 24x^2 - 8x^3 + x^4 = 10000 * (x-2)^2

Simplify the equation:

x^4 - 8x^3 + 24x^2 - 32x + 16 = 10000(x^2 - 4x + 4)

x^4 - 8x^3 + 24x^2 - 32x + 16 = 10000x^2 - 40000x + 40000

Rearrange the equation and combine like terms:

x^4 - 8x^3 + (24 - 10000)x^2 + (40000 - 32)x + (16 - 40000) = 0

x^4 - 8x^3 + 99976x^2 - 39968x - 39984 = 0

Unfortunately, this equation does not have an exact algebraic solution. You would need to solve it numerically using approximation methods or numerical solvers.