Question
Find the value of:$\frac{\log \sqrt{27}+\log 8+\log \sqrt{1000}}{\log 120}$
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Answer
$\frac{\log \sqrt{27}+\log 8+\log \sqrt{1000}}{\log 120} $
$ =\frac{\log (27)^{\frac{1}{2}}+\log 2^3+\log 1000^{\frac{1}{2}}}{\log \left(3 \times 2^2 \times 10\right)} $
$ =\frac{\log (3)^{3 \times \frac{1}{2}}+\log 2^3+\log (10)^{3 \times \frac{1}{2}}}{\log 3+\log 2^{\wedge} 2+\log 10} $
$ =\frac{\frac{3}{2} \log 3+3 \log 2+\frac{3}{2} \log (10)}{\log 3+2 \log 2+\log 10} $
$ =\frac{\frac{3}{2} \log 3+\frac{3}{2}(2 \log 2)+\frac{3}{2}(1)}{\log 3+2 \log 2+1} $
$ =\frac{\frac{3}{2}[\log 3+2 \log 2+1]}{\log 3+2 \log 2+1}$
$ =\frac{3}{2} .$
$ =\frac{\log (27)^{\frac{1}{2}}+\log 2^3+\log 1000^{\frac{1}{2}}}{\log \left(3 \times 2^2 \times 10\right)} $
$ =\frac{\log (3)^{3 \times \frac{1}{2}}+\log 2^3+\log (10)^{3 \times \frac{1}{2}}}{\log 3+\log 2^{\wedge} 2+\log 10} $
$ =\frac{\frac{3}{2} \log 3+3 \log 2+\frac{3}{2} \log (10)}{\log 3+2 \log 2+\log 10} $
$ =\frac{\frac{3}{2} \log 3+\frac{3}{2}(2 \log 2)+\frac{3}{2}(1)}{\log 3+2 \log 2+1} $
$ =\frac{\frac{3}{2}[\log 3+2 \log 2+1]}{\log 3+2 \log 2+1}$
$ =\frac{3}{2} .$
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