Write the expression as a single logarithm. Express powers as factors. \[ \log _{7} \sqrt{x}-\log _{7} x^{7} \]

To combine the expression \( \log_{7} \sqrt{x}-\log_{7} x^{7} \) into a single logarithm, we can use the properties of logarithms. First, recall that \( \sqrt{x} \) can be rewritten as \( x^{1/2} \). Thus, we have: \[ \log_{7} \sqrt{x} = \log_{7} x^{1/2} \] Now we apply the power rule of logarithms: \[ \log_{7} x^{1/2} = \frac{1}{2} \log_{7} x \] Next, substitute this back into the original expression: \[ \frac{1}{2} \log_{7} x - \log_{7} x^{7} \] Using the power rule again, we rewrite \( \log_{7} x^{7} \) as: \[ \log_{7} x^{7} = 7 \log_{7} x \] Now substituting back gives: \[ \frac{1}{2} \log_{7} x - 7 \log_{7} x \] To combine these, we find a common denominator, which is 2: \[ \frac{1}{2} \log_{7} x - \frac{14}{2} \log_{7} x = \frac{1 - 14}{2} \log_{7} x = \frac{-13}{2} \log_{7} x \] Finally, we can express this as: \[ \log_{7} x^{-\frac{13}{2}} \] Thus, the expression as a single logarithm is: \[ \log_{7} x^{-\frac{13}{2}} \]