The value of $$5log1/5$$⁡$$(1/2)+log2$$⁡$$47+3+log1/2$$⁡$$110+221$$ is

$$5log1/5$$⁡$$(1/2)+log2$$⁡$$47+3+log1/2$$⁡$$110+221=2+log2$$⁡$$47+32(7+3)2=2+log2$$⁡$$16=6$$

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Question

$The value of 5^{\log_{1 / 5} (1 / 2)} + \log_{\sqrt{2}} \frac{4}{\sqrt{7} + \sqrt{3}} + \log_{1 / 2} \frac{1}{10 + 2 \sqrt{21}} is$

Moderate

Solution

$\begin{array}{l} 5^{\log_{1 / 5} (1 / 2)} + \log_{\sqrt{2}} \frac{4}{\sqrt{7} + \sqrt{3}} + \log_{1 / 2} \frac{1}{10 + 2 \sqrt{21}} \\ = 2 + \log_{2} \{{(\frac{4}{\sqrt{7} + \sqrt{3}})}^{2} (\sqrt{7} + \sqrt{3})^{2}\} \\ = 2 + \log_{2} 16 = 6 \end{array}$

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