If `log_10 343 = 2.5353` then the least positive integer 'n' such that `7^n>10^5` is

Options

A. 1
B. 6
C. 5
D. 4

Answer

To solve the problem, we need to find the least positive integer \( n \) such that \( 7^n > 10^5 \). We are given that \( \log_{10} 343 = 2.5353 \).

### Step-by-Step Solution:

1. **Express 343 in terms of base 7**:
\[ 343 = 7^3 \]
Therefore, we can write:
\[ \log_{10} 343 = \log_{10} (7^3) = 3 \log_{10} 7 \]

2. **Relate \( \log_{10} 7 \) to the given value**:
From the equation above, we have:
\[ 3 \log_{10} 7 = 2.5353 \]
To find \( \log_{10} 7 \), we divide both sides by 3:
\[ \log_{10} 7 = \frac{2.5353}{3} \approx 0.8451 \]

3. **Set up the inequality**:
We need to find \( n \) such that:
\[ 7^n > 10^5 \]
Taking logarithm (base 10) on both sides gives:
\[ \log_{10} (7^n) > \log_{10} (10^5) \]
This simplifies to:
\[ n \log_{10} 7 > 5 \]

4. **Substitute the value of \( \log_{10} 7 \)**:
Substitute \( \log_{10} 7 \) into the inequality:
\[ n \cdot 0.8451 > 5 \]

5. **Solve for \( n \)**:
To isolate \( n \), divide both sides by \( 0.8451 \):
\[ n > \frac{5}{0.8451} \approx 5.917 \]

6. **Determine the least positive integer \( n \)**:
Since \( n \) must be a positive integer, we round up \( 5.917 \) to the next whole number:
\[ n = 6 \]

### Conclusion:
The least positive integer \( n \) such that \( 7^n > 10^5 \) is \( \boxed{6} \).