[ [3] x-2 (2 x+1)^4 (x+4) 2 x+4 ]^2 - Vidyadip

Question

$\ln \left[\frac{\sqrt[3]{x-2}(2 x+1)^4}{(x+4) \sqrt{2 x+4}}\right]^2$

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Answer

$ \ln \left[\frac{\sqrt[3]{x-2}(2 x+1)^4}{(x+4) \sqrt{2 x+4}}\right]^2$
$=2 \ln \left[\frac{\sqrt[3]{x-2}(2 x+1)^4}{(x+4) \sqrt{2 x+4}}\right] \quad \ldots\left[\log \mathrm{m}^{\mathrm{n}}=\mathrm{n} \log \mathrm{m}\right]$
$=2\left[\ln \sqrt[3]{x-2}(2 x+1)^4-\ln (x+4) \sqrt{2 x+4}\right]$
$=2\left[\ln \sqrt[3]{x-2}+\ln (2 x+1)^4-(\ln (x+4)+\ln \sqrt{2 x+4})\right] $
$=2\left[\frac{1}{3} \ln (x-2)+4\right. \ln (2 x+1) \left.-\ln (x+4)-\frac{1}{2} \ln (2 x+4)\right] $

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