To evaluate \( \log_{\sqrt{100}} 3 \), we can start by expressing \(\sqrt{100}\) in simpler terms.
We know that: \[ \sqrt{100} = 10 \]
Thus, the expression can be rewritten as: \[ \log_{10} 3 \]
Now, the value of \( \log_{10} 3 \) represents the exponent to which the base 10 must be raised to produce the number 3. While we cannot express this as an exact number without a calculator, we can use known approximations.
The approximate value of \( \log_{10} 3 \) is around \( 0.4771 \).
So, we conclude: \[ \log_{\sqrt{100}} 3 \approx 0.4771 \]
If you need a more precise numeric value, you can use a scientific calculator to find it. However, the exact evaluation is presented as \( \log_{10} 3 \).