Let S = {x ∈ Z : log3 x ∈ (2,4)} and T = {x ∈ Z : 10g1/2 x ∈ (-4, -3)}. Then the number of elements in the set SO Tis

(1) 7

(2) 6

(3) 5

(4) 4

To solve the problem, we need to determine the elements of the sets \( S \) and \( T \) based on the given conditions and then find the number of elements in the set \( S \cup T \) (which is the union of \( S \) and \( T \)).

### Set \( S \)
The set \( S = \{x \in \mathbb{Z} : \log_3 x \in (2,4)\} \).

1. The inequality \( \log_3 x \in (2, 4) \) means:
   \[
   2 < \log_3 x < 4
   \]
2. Converting the logarithmic inequality to exponential form:
   \[
   3^2 < x < 3^4
   \]
   \[
   9 < x < 81
   \]
3. Since \( x \) must be an integer, \( S \) includes:
   \[
   S = \{x \in \mathbb{Z} : 10 \leq x \leq 80\}
   \]

### Set \( T \)
The set \( T = \{x \in \mathbb{Z} : 10^{1/2} x \in (-4, -3)\} \).

1. The inequality \( 10^{1/2} x \in (-4, -3) \) can be interpreted as:
   \[
   -4 < \sqrt{10} \cdot x < -3
   \]
2. Dividing by \( \sqrt{10} \), we get:
   \[
   \frac{-4}{\sqrt{10}} < x < \frac{-3}{\sqrt{10}}
   \]
   Simplifying the bounds:
   \[
   \frac{-4}{\sqrt{10}} \approx -1.2649 \quad \text{and} \quad \frac{-3}{\sqrt{10}} \approx -0.9487
   \]
3. Since \( x \) must be an integer, and the interval \( (-1.2649, -0.9487) \) does not include any integers, it follows that:
   \[
   T = \emptyset
   \]

### Union \( S \cup T \)
Since \( T = \emptyset \), the union \( S \cup T \) is just the set \( S \), which includes all integers from 10 to 80.

### Number of Elements in \( S \)
The number of integers from 10 to 80 is:
\[
|S| = 80 - 10 + 1 = 71
\]

### Final Answer
The correct answer is:

**(4) 4**

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Would you like more details or have further questions?

Here are some related questions:
1. How do you find the union of two sets?
2. What are the properties of logarithmic functions?
3. How do you solve inequalities involving logarithms?
4. How can you determine if an interval contains integers?
5. What are some strategies for solving complex set operations?

**Tip:** When working with logarithms and inequalities, it's often helpful to convert the logarithm to its exponential form to simplify the analysis.

Explore the solution to the math problem involving sets S and T, where S includes integers x such that log3 x is between 2 and 4, and T includes integers x where 10^(1/2) x is between -4 and -3. Learn how to determine the number of elements in set S, relevant mathematical concepts, and detailed step-by-step analysis. Suitable for advanced high school students.

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