Given `log_(10)` 343 = 2.5353, the least integer n such that ` 7^(n) gt 10^(10)` is

Options

A. 10
B. 11
C. 12
D. 13

Answer

To solve the problem, we need to find the least integer \( n \) such that \( 7^n > 10^{10} \). We can do this by taking logarithms of both sides.

### Step-by-step Solution:

1. **Take logarithm of both sides**:
\[ \log_{10}(7^n) > \log_{10}(10^{10}) \]

2. **Apply the logarithm power rule**:
Using the property of logarithms that states \( \log_b(a^c) = c \cdot \log_b(a) \), we can rewrite the left side:
\[ n \cdot \log_{10}(7) > 10 \]

3. **Isolate \( n \)**:
To isolate \( n \), divide both sides by \( \log_{10}(7) \):
\[ n > \frac{10}{\log_{10}(7)} \]

4. **Calculate \( \log_{10}(7) \)**:
We can find \( \log_{10}(7) \) using the given information. Since \( \log_{10}(343) = 2.5353 \) and \( 343 = 7^3 \), we can find \( \log_{10}(7) \):
\[ \log_{10}(7) = \frac{\log_{10}(343)}{3} = \frac{2.5353}{3} \approx 0.8451 \]

5. **Substitute \( \log_{10}(7) \) back into the inequality**:
Now we substitute this value back into our inequality:
\[ n > \frac{10}{0.8451} \approx 11.83 \]

6. **Find the least integer \( n \)**:
Since \( n \) must be an integer, we take the ceiling of \( 11.83 \):
\[ n = 12 \]

### Conclusion:
The least integer \( n \) such that \( 7^n > 10^{10} \) is \( \boxed{12} \).