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

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

ОДЗ:

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

x ∈ (4; +∞)

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

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

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

$x^2 - 6x + 8 = 8\\x (x-6) = 0\\\\\ \left[\begin{array}{ccc}x = 0\\x = 6\\x4 \right\end{array}\right$

ответ: 6.