Alg9.4.43

Find the value of $\log_{\sqrt{2}} \sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}\,$

$\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}=\sqrt{2}\cdot\sqrt[4]{2}\cdot\sqrt[8]{2}\cdot\sqrt[16]{2}\cdot\sqrt[32]{2}\,$

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

$2^{\frac{31}{32}}\,$

Now

$\log_{\sqrt{2}}(2^{\frac{31}{32}})\,$

$\log_{\sqrt{2}}(2^{\frac{31}{32}})=\frac{31}{32}\cdot2\,$

Hence the solution is 31/16.