Question
Find the value of:$\frac{\log \sqrt{125}-\log \sqrt{27}-\log \sqrt{8}}{\log 6-\log 5}$
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Answer
$\frac{\log \sqrt{125}-\log \sqrt{27}-\log \sqrt{8}}{\log 6-\log 5}$
$=\frac{\log (125)^{\frac{1}{2}}-\log (27)^{\frac{1}{2}}-\log (8)^{\frac{1}{2}}}{\log 6-\log 5}$
$=\frac{\log (5)^{3 \times \frac{1}{2}}-\log (3)^{3 \times \frac{1}{2}}-\log (2)^{3 \times \frac{1}{2}}}{\log 6-\log 5}$
$=\frac{\frac{3}{2} \log (5)-\frac{3}{2} \log (3)-\frac{3}{2} \log (2)}{\log (2 \times 3)-\log 5}$
$=\frac{\frac{3}{2}[\log (5)-\log (3)-\log (2)]}{\log 2+\log 3-\log 5}$
$=\frac{\frac{3}{2}[\log (5)-\log (3)-\log (2)]}{-[\log 5-\log 3-\log 2]}$
$=-\frac{3}{2} .$
$=\frac{\log (125)^{\frac{1}{2}}-\log (27)^{\frac{1}{2}}-\log (8)^{\frac{1}{2}}}{\log 6-\log 5}$
$=\frac{\log (5)^{3 \times \frac{1}{2}}-\log (3)^{3 \times \frac{1}{2}}-\log (2)^{3 \times \frac{1}{2}}}{\log 6-\log 5}$
$=\frac{\frac{3}{2} \log (5)-\frac{3}{2} \log (3)-\frac{3}{2} \log (2)}{\log (2 \times 3)-\log 5}$
$=\frac{\frac{3}{2}[\log (5)-\log (3)-\log (2)]}{\log 2+\log 3-\log 5}$
$=\frac{\frac{3}{2}[\log (5)-\log (3)-\log (2)]}{-[\log 5-\log 3-\log 2]}$
$=-\frac{3}{2} .$
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