log₁₀₀√10

$$log_{100} \sqrt{10} = log_{10^2} 10^{\frac{1}{2}} = \frac{1}{2} log_{10^2} 10 = \frac{1}{2} \cdot \frac{1}{2} log_{10} 10 = \frac{1}{4} \cdot 1 = \frac{1}{4}$$