Solve the system of inequalities $$\left\{ \begin{array} { l } \log _ { 3 } | x - 3 | < 4 \\ \log _ { 2 } x + \log _ { 2 } ( x - 2 ) \geqq 3 \end{array} \right.$$ and find the number of integers $x$ that satisfy it. [3 points]

System of inequalities $$\left\{ \begin{array} { l } \log _ { 3 } | x - 3 | < 4 \\ \log _ { 2 } x + \log _ { 2 } ( x - 2 ) \geqq 3 \end{array} \right.$$ Find the number of integers $x$ that satisfy the system. [3 points]

Find the maximum natural number $x$ that satisfies the inequality $\left( \log _ { 3 } x \right) \left( \log _ { 3 } 3 x \right) \leqq 20$. [3 points]

Find the number of natural numbers $x$ that satisfy the logarithmic inequality $$\log _ { 2 } x \leqq \log _ { 4 } ( 12 x + 28 )$$ [3 points]

The domain of the function $f ( \mathrm { x } ) = \log _ { 2 } \left( \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 3 \right)$ is
(A) $[ - 3,1 ]$
(B) $( - 3,1 )$
(C) $( - \infty , - 3 ] \cup [ 1 , + \infty )$
(D) $( - \infty , - 3 ) \cup ( 1 , + \infty )$

$$0 \leq \log_{2}(x-5) \leq 2$$
How many integers $x$ satisfy these inequalities?
A) 2
B) 3
C) 4
D) 5
E) 6