Alg9.4.42

Find the value of $\log_{\sqrt{a}} \sqrt{a\sqrt{a\sqrt{a\sqrt{a\sqrt{a}}}}}\,$

$\sqrt{a\sqrt{a\sqrt{a\sqrt{a\sqrt{a}}}}}=\sqrt{a}\cdot\sqrt[4]{a}\cdot\sqrt[8]{a}\cdot\sqrt[16]{a}\cdot\sqrt[32]{a}\,$

$a^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}}\,$

$a^{\frac{31}{32}}\,$

Now

$\log_{\sqrt{a}}(a^{\frac{31}{32}})\,$

$\log_{\sqrt{a}}(a^{\frac{31}{32}})=\frac{31}{32}\cdot2\,$

Hence the solution is 31/16.