To evaluate \( \log_{1,000}\sqrt{4} \), we can first rewrite it using properties of logarithms.
- Rewrite \( \sqrt{4} \) as \( 4^{1/2} \).
- Thus, \( \log_{1,000}\sqrt{4} = \log_{1,000}\left(4^{1/2}\right) \).
Using the logarithmic identity \( \log_b(a^c) = c \cdot \log_b(a) \), we can simplify:
\[ \log_{1,000}\left(4^{1/2}\right) = \frac{1}{2} \cdot \log_{1,000}(4) \]
Now, we need to find \( \log_{1,000}(4) \). Since \( 1,000 = 10^3 \), we can use the change of base formula:
\[ \log_{1,000}(4) = \frac{\log_{10}(4)}{\log_{10}(1,000)} = \frac{\log_{10}(4)}{3} \]
Putting it together:
\[ \log_{1,000}\sqrt{4} = \frac{1}{2} \cdot \frac{\log_{10}(4)}{3} = \frac{\log_{10}(4)}{6} \]
Now we know \( 4 \) can be expressed as \( 2^2 \):
\[ \log_{10}(4) = \log_{10}(2^2) = 2\log_{10}(2) \]
Thus, \[ \frac{\log_{10}(4)}{6} = \frac{2\log_{10}(2)}{6} = \frac{\log_{10}(2)}{3} \]
Now, using properties of logarithms, since \( 2 = 10^{\log_{10}(2)} \):
To evaluate \( \log_{1000}(4) \), we can also write: \[ \log_{1,000}(4) = \log_{1,000}(10^{2}) = \frac{2}{3} \]
Thus: \[ \log_{1000}\sqrt{4} = \frac{1}{2} \cdot \frac{2}{3} = \frac{1}{3} \]
So, the final result is: \[ \log_{1000}\sqrt{4} = \frac{1}{3} \]
From the options provided, it appears none of them match this result directly. However, if \( \log_{1000}4 = \frac{2}{3} \) then \( \frac{1}{2} \cdot \log_{1000}(4) \rightarrow \frac{1}{2}\left(\frac{2}{3}\right)= \frac{1}{3}.\)
Please verify your options to confirm or provide additional details if needed.