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To find the value of \(\log _{\sqrt{3}}81\), we can follow these steps: Step 1: Express 81 as a power of 3 We know that:\(81=3\times 3\times 3\times 3=3^{4}\)Step 2: Express 3 as a power of \(\sqrt{3}\) Since \(3=(\sqrt{3})^{2}\), we can substitute this into the equation for 81:\(81=((\sqrt{3})^{2})^{4}=(\sqrt{3})^{8}\)Step 3: Apply the Logarithm Now, substitute \((\sqrt{3})^{8}\) back into the original expression:\(\log _{\sqrt{3}}81=\log _{\sqrt{3}}(\sqrt{3})^{8}\)Using the power rule of logarithms, \(\log _{b}(M^{k})=k\log _{b}M\):\(8=8\log _{\sqrt{3}}\sqrt{3}\)Since \(\log _{b}b=1\):\(8\times 1=8\)Result: The value of \(\log _{\sqrt{3}}81\) is 8

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