Logarithms again......

 Simplify 

$\dfrac{\log\sqrt8+\log\sqrt{125}-\log\sqrt{27}}{\log2+\log\sqrt[3]{5}-\log\sqrt[3]{12}}$

$\begin{array}{rcll} && \dfrac{\log\sqrt8+\log\sqrt{125}-\log\sqrt{27}}{\log2+\log\sqrt[3]{5}-\log\sqrt[3]{12}}\\\\ &=& \dfrac{\log{\sqrt{2^3}}+\log{\sqrt{5^3}}-\log{\sqrt{3^3}} }{\log(2)+\log{\sqrt[3]{5}} - \log{\sqrt[3]{3\cdot2^2}} }\\\\ &=& \dfrac{\log{\sqrt{2^3}}+\log{\sqrt{5^3}}-\log{\sqrt{3^3}} }{\log(2)+\log{\sqrt[3]{5}} - \log(\sqrt[3]{3}\cdot \sqrt[3]{2^2}) }\\\\ &=& \dfrac{\log{\sqrt{2^3}}+\log{\sqrt{5^3}}-\log{\sqrt{3^3}} }{\log(2)+\log{\sqrt[3]{5}} -\log\sqrt[3]{3} -\log\sqrt[3]{2^2} }\\\\ &=& \dfrac{\log{\sqrt{2^3}}+\log{\sqrt{5^3}}-\log{\sqrt{3^3}} }{\log(2)+\log{\sqrt[3]{5}} -\log\sqrt[3]{3} -\log(2^\frac23 ) }\\\\ &=& \dfrac{\log{\sqrt{2^3}}+\log{\sqrt{5^3}}-\log{\sqrt{3^3}} }{\log(2)+\log{\sqrt[3]{5}} -\log\sqrt[3]{3} -\frac23\cdot \log(2) }\\\\ &=& \dfrac{\log{\sqrt{2^3}}+\log{\sqrt{5^3}}-\log{\sqrt{3^3}} }{\log(2)-\frac23\cdot \log(2) +\log{\sqrt[3]{5}} -\log\sqrt[3]{3} }\\\\ &=& \dfrac{\log{\sqrt{2^3}}+\log{\sqrt{5^3}}-\log{\sqrt{3^3}} }{\frac33\cdot\log(2)-\frac23\cdot \log(2) +\log{\sqrt[3]{5}} -\log\sqrt[3]{3} }\\\\ &=& \dfrac{\log{\sqrt{2^3}}+\log{\sqrt{5^3}}-\log{\sqrt{3^3}} }{\frac13\cdot\log(2) +\log{\sqrt[3]{5}} -\log\sqrt[3]{3} }\\\\ &=& \dfrac{\log{\sqrt{2^3}}+\log{\sqrt{5^3}}-\log{\sqrt{3^3}} }{\frac13\cdot\log(2) +\frac13\cdot\log(5) -\frac13\cdot\log(3) }\\\\ &=& \dfrac{ \frac32\cdot\log(2)+\frac32\cdot\log(5)-\frac32\cdot\log(3) }{\frac13\cdot\log(2) +\frac13\cdot\log(5) -\frac13\cdot\log(3) }\\\\ &=& \dfrac{\frac32}{\frac13} \left( \dfrac{ \log(2)+\log(5)-\log(3) }{\log(2) +\log(5) -\log(3) } \right) \\\\ &=& \dfrac{\frac32}{\frac13} \\\\ &=& \frac32 \cdot \frac31 \\\\ &=& \frac92 \\\\ &=& 4.5 \end{array}$

[ log√8 + log √125  - log √27 ]  /  [ log 2 + log ³√5  -  log ³√12 ] =

[ log√8 + log √125  - log √27 ]  / log ³√8 + log ³√5 - log³√12 ] =

[ (1/2) log 8 + (1/2) log 125  -  (1/2)log 27 ] / [ (1/3) log 8  + (1/3) log 5 - (1/3) log12] =

[(1/2) log 2³ + (1/2) log5³ - (1/2) log 3³ ] / [ (1/3) log 2³ + (1/3) log 5 - (1/3) log12 ] =

 (3/2)[ log 2 + log 5 - log 3] / [ (1/3) [ log2³ + log 5 - log 12] ] =

(3/2) [ log ( [2 * 5]  / 3 )  /  [ (1/3) [log [2³ * 5 / 12 ]  )  =

(3/2) log (10/3) /  [(1/3) log (40/12) ]  =

(3/2) log (10/3) / [ (1/3) log (10/3) ] =

(3/2)  / (1/3)  = 

(3/2) * (3/1)  =

9/2  =

4.5