Question 109

The value of $$\text{log}_{7} \text{log}_{7} \sqrt{7(\sqrt{7\sqrt{7}})}$$


$$\text{log}_{7} \text{log}_{7} \sqrt{7(\sqrt{7\sqrt{7}})}$$ 
= $$\text{log}_{7} [\frac{1}{2} (\text{log}_{7} 7+\text{log}_{7} \sqrt{7(\sqrt{7})})]$$
= $$\text{log}_{7} [\frac{1}{2} (1 +\frac{1}{2}\text{log}_{7}{7(\sqrt{7})})]$$
= $$\text{log}_{7} [\frac{1}{2} (1 +\frac{1}{2}(1+1/2))]$$
= $$\text{log}_{7}\frac{7}{8}$$
= $$\text{log}_{7}7 - \text{log}_{7}8$$
= $$1-3 \text{log}_{7}2$$
Hence, option D is the correct answer.

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