Question

1/2 log2 (x-2)^2+1/3 log2(x-4)^3=3​

Answer 1

log(a^k) b^b = b/k log(a) b

log a + log b = log ab

1/2 log2 (x-2)^2+1/3 log2(x-4)^3=3​

одз x ≠ 2   x>4

1/2 *2  log2 (x-2)+1/3 *3  log2(x-4) = 3​

log2 (x-2) + log2(x-4) = 3​

log2 (x-2)(x-4) = 3​

(x-2)(x-4) = 8

​x^2 - 2x - 4x + 8 = 8

​x(x - 6) = 0

x = 0 не проходит по одз

x = 6

Answer 2

Ответ:

$frac{1}{2} log_2(x-2)^2 + frac{1}{3} log_2(x-4)^3 = 3\\frac{2}{2} log_2(x-2) + frac{3}{3} log_2(x-4) = 3$

ОДЗ:

$(x-2)>0\(x-4) > 0$

x ∈ (4; +∞)

--------------------------

$log_2((x-2)*(x-4)) = 3$

$log_2(x^2 - 6x + 8) = log_22^3$

$x^2 - 6x + 8 = 8\x (x-6) = 0\\ left[begin{array}{ccc}x = 0\x = 6\x>4 rightend{array}right$

Ответ: 6.