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How many values ​​of n satisfy the double inequality?

To solve the double inequality \( 4 < 2^n < 1024 \) and find how many values of \( n \) satisfy this inequality, follow these steps:

### Step 1: Solve the lower bound
We need to solve \( 4 < 2^n \).

Since \( 2^n \) is an exponential function, we can take logarithms, or simply observe that:
\[
2^n > 4 = 2^2
\]
Thus, \( n > 2 \).

### Step 2: Solve the upper bound
Now solve \( 2^n < 1024 \).

Again, using the properties of powers of 2:
\[
1024 = 2^{10}
\]
So, \( n < 10 \).

### Step 3: Combine the results
We now have:
\[
2 < n < 10
\]

### Step 4: Identify the integer values of \( n \)
The integer values that satisfy this inequality are \( n = 3, 4, 5, 6, 7, 8, 9 \).

### Final Answer
There are 7 integer values of \( n \) that satisfy the inequality.

Would you like more details or have any further questions?

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Here are some related questions you might be interested in:

1. What is the general method for solving double inequalities?
2. How does logarithmic manipulation help in solving exponential inequalities?
3. What is the significance of base 2 in this inequality?
4. Can the inequality be solved graphically? If so, how?
5. How does changing the bounds (e.g., 4 and 1024) affect the solution?
6. What if the inequality was \( 2^n \leq 1024 \)? How would the solution change?
7. How does solving \( a < x < b \) differ from solving \( ax < b \)?
8. What are the implications of exponential growth when solving inequalities?

**Tip:** When dealing with inequalities involving powers, remember that the properties of exponents and logarithms are key tools in simplifying and solving the problem.

Learn how to solve the exponential inequality 4 < 2^n < 1024 with detailed steps. Discover the significance of exponential functions and logarithms in solving such inequalities. Suitable for advanced high school students.

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