The value of x for which the unit digits of `(2357)^(log_10 x)` and `(5723)^(x)` is same for `x > 1`.

Options

A. 10
B. 100
C. 1000
D. none of these

Answer

To find the value of \( x \) for which the unit digits of \( (2357)^{\log_{10} x} \) and \( (5723)^{x} \) are the same for \( x > 1 \), we will follow these steps:

### Step 1: Identify the unit digits of the bases
- The unit digit of \( 2357 \) is \( 7 \).
- The unit digit of \( 5723 \) is \( 3 \).

### Step 2: Determine the unit digit pattern for \( 7^n \)
The unit digits of powers of \( 7 \) cycle every 4:
- \( 7^1 \equiv 7 \)
- \( 7^2 \equiv 9 \)
- \( 7^3 \equiv 3 \)
- \( 7^4 \equiv 1 \)
- (Then it repeats)

### Step 3: Determine the unit digit pattern for \( 3^n \)
The unit digits of powers of \( 3 \) also cycle every 4:
- \( 3^1 \equiv 3 \)
- \( 3^2 \equiv 9 \)
- \( 3^3 \equiv 7 \)
- \( 3^4 \equiv 1 \)
- (Then it repeats)

### Step 4: Find the unit digit of \( (2357)^{\log_{10} x} \)
We need to find \( \log_{10} x \mod 4 \) to determine which unit digit of \( 7 \) we will get:
- If \( \log_{10} x \equiv 1 \mod 4 \), unit digit is \( 7 \)
- If \( \log_{10} x \equiv 2 \mod 4 \), unit digit is \( 9 \)
- If \( \log_{10} x \equiv 3 \mod 4 \), unit digit is \( 3 \)
- If \( \log_{10} x \equiv 0 \mod 4 \), unit digit is \( 1 \)

### Step 5: Find the unit digit of \( (5723)^{x} \)
We need to find \( x \mod 4 \) to determine which unit digit of \( 3 \) we will get:
- If \( x \equiv 1 \mod 4 \), unit digit is \( 3 \)
- If \( x \equiv 2 \mod 4 \), unit digit is \( 9 \)
- If \( x \equiv 3 \mod 4 \), unit digit is \( 7 \)
- If \( x \equiv 0 \mod 4 \), unit digit is \( 1 \)

### Step 6: Set up equations based on the unit digits
We need to find \( x \) such that the unit digits match:
1. If \( \log_{10} x \equiv 1 \mod 4 \) and \( x \equiv 1 \mod 4 \):
- This gives \( 7 \) from \( 7^{\log_{10} x} \) and \( 3^{x} \).
2. If \( \log_{10} x \equiv 2 \mod 4 \) and \( x \equiv 2 \mod 4 \):
- This gives \( 9 \) from both.
3. If \( \log_{10} x \equiv 3 \mod 4 \) and \( x \equiv 3 \mod 4 \):
- This gives \( 3 \) from both.
4. If \( \log_{10} x \equiv 0 \mod 4 \) and \( x \equiv 0 \mod 4 \):
- This gives \( 1 \) from both.

### Step 7: Solve for \( x \)
To satisfy the conditions, we can try specific values of \( x \):
- If \( x = 10 \):
- \( \log_{10} 10 = 1 \) (which is \( 1 \mod 4 \)), so \( 7 \) from \( (2357)^{1} \).
- \( 10 \mod 4 = 2 \), so \( 9 \) from \( (5723)^{10} \).
- Not equal.
- If \( x = 100 \):
- \( \log_{10} 100 = 2 \) (which is \( 2 \mod 4 \)), so \( 9 \) from \( (2357)^{2} \).
- \( 100 \mod 4 = 0 \), so \( 1 \) from \( (5723)^{100} \).
- Not equal.
- If \( x = 1000 \):
- \( \log_{10} 1000 = 3 \) (which is \( 3 \mod 4 \)), so \( 3 \) from \( (2357)^{3} \).
- \( 1000 \mod 4 = 0 \), so \( 1 \) from \( (5723)^{1000} \).
- Not equal.
- If \( x = 10000 \):
- \( \log_{10} 10000 = 4 \) (which is \( 0 \mod 4 \)), so \( 1 \) from \( (2357)^{4} \).
- \( 10000 \mod 4 = 0 \), so \( 1 \) from \( (5723)^{10000} \).
- Equal.

### Final Answer
The value of \( x \) for which the unit digits are the same is \( x = 10000 \).