Log Reciprocal Base E8753E

1. **Problem statement:** Evaluate the expression $$\log_{\sqrt{a}} \frac{1}{y}$$. 2. **Recall the logarithm change of base and properties:** - The logarithm of a reciprocal: $$\log_b \frac{1}{x} = -\log_b x$$. - The base $$\sqrt{a}$$ can be rewritten as $$a^{1/2}$$. 3. **Rewrite the expression using these properties:** $$\log_{\sqrt{a}} \frac{1}{y} = -\log_{\sqrt{a}} y$$. 4. **Change the base from $$\sqrt{a}$$ to $$a$$:** $$\log_{\sqrt{a}} y = \frac{\log_a y}{\log_a \sqrt{a}} = \frac{\log_a y}{\log_a a^{1/2}} = \frac{\log_a y}{\frac{1}{2}} = 2 \log_a y$$. 5. **Substitute back:** $$-\log_{\sqrt{a}} y = -2 \log_a y$$. **Final answer:** $$\log_{\sqrt{a}} \frac{1}{y} = -2 \log_a y$$.