How to simplify $\log_7\log_7\sqrt{7\sqrt{7\sqrt{7}}}$

Question

As said in the question, I have this expression -

$$\log_7\log_7\sqrt{7\sqrt{7\sqrt{7}}}.$$

I have to find the value of this expression. However, I don't seem to understand how to proceed with simplifying $\sqrt{7\sqrt{7\sqrt{7}}}$.

I've tried looking at this, but it doesn't help me, since here there's another number inside the $\sqrt{}$

Answer 1

Hint. Note that $$\sqrt{7\sqrt{7\sqrt{7}}}=\sqrt{7\sqrt{7^{3/2}}}=\sqrt{7^{7/4}}=7^{7/8}.$$ It remains to simplify $$\log_7(\log_7(7^{7/8})).$$

Answer 2

$$\sqrt{7\sqrt{7\sqrt{7}}}=\sqrt{7}\cdot \sqrt{\sqrt{7}}\cdot \sqrt{\sqrt{\sqrt{7}}}=7^{\frac 12+\frac 14+\frac 18}=7^\frac 78$$

Answer 3

$$\log _7\left(\log _7\left(\sqrt{7\sqrt{7\sqrt{7}}}\right)\right)=1-3\log _7\left(2\right)$$

In fact

$$\log _7\left(\log _7\left(\sqrt{7\sqrt{7\sqrt{7}}}\right)\right)=\log _7\left(\log _7\left(\left(7\sqrt{7\sqrt{7}}\right)^{\frac{1}{2}}\right)\right)=$$

$$=\log _7 \left(\frac{1}{2}\log _7\left(7\sqrt{7\sqrt{7}}\right)\right)=\log _7\left(\frac{1}{2}\log _7\left(7\cdot \:7^{\frac{3}{4}}\right)\right)=\log _7\left(\frac{1}{2}\cdot \frac{7}{4}\right)=\log _7\left(\frac{7}{8}\right)=$$

$$=\log _7\left(7\right)-\log _7\left(8\right)=1-\log _7\left(8\right)=1-3\log _7\left(2\right)$$

Answer 4

$$y=\log_{7}(\log_{7}(\sqrt{7\sqrt{7\sqrt{7}}}))=\log_{7}(\frac{1}{2}\log_{7}(7\sqrt{7\sqrt{7}})$$ now $$7\sqrt{7\sqrt{7}}=7(7.7^{\frac{1}{2}})^{\frac{1}{2}}=7.7^{\frac{3}{4}}=7^{\frac{7}{4}}$$ $$\therefore y=log_{7}(\frac{7}{8})=log_{7}(7)-log_{7}(8)=1-log_{7}(8)$$