1. The problem is to simplify or evaluate $\log_p \sqrt{x}$. 2. Recall that the square root of $x$ can be written as an exponent: $\sqrt{x} = x^{\frac{1}{2}}$. 3. Using the logarithm power rule, $\log_p \left(x^{\frac{1}{2}}\right) = \frac{1}{2} \log_p x$. 4. Therefore, the simplified expression is: $$\log_p \sqrt{x} = \frac{1}{2} \log_p x$$ This shows how the logarithm of a square root translates into a fraction of the logarithm of the base argument.