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I'm preparing for my further studies (last year of high school, preparing so I can try and join the academy that I want), and just solving problems. Got stuck on this one:

What is the value of: $$log_4log_3log_28 + log_{\sqrt{7}+1}(8+2\sqrt{7})+log_{\sqrt[3]{7}}7\sqrt{7}$$

This is what I got so far:

$log_4log_3log_28 = log_4log_33=log_41=0$

$log_{\sqrt[3]{7}}7\sqrt{7}=log_{7^{1\over3}}(7*7^{1\over2})=3log_77^{3\over2}=3*{3\over2}log_77={9\over2}$

So

$log_4log_3log_28 + log_{\sqrt{7}+1}(8+2\sqrt{7})+log_{\sqrt[3]{7}}7\sqrt{7} \\= 0 + log_{\sqrt{7}+1}(8+2\sqrt{7}) + {9\over2}\\={9\over2}+log_{\sqrt{7}+1}(8+2\sqrt{7})$

I'm lost at what to do with $log_{\sqrt{7}+1}(8+2\sqrt{7})$

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Hint: $$8+2\sqrt{7}=(\sqrt{7}+1)^2$$

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