120125

[120125] $A={{\log }_{2}}3\cdot {{\log }_{3}}4\cdot {{\log }_{4}}5\cdot {{\log }_{5}}6\cdot {{\log }_{6}}7\cdot {{\log }_{7}}8$

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$A={{\log }_{2}}3\cdot {{\log }_{3}}4\cdot {{\log }_{4}}5\cdot {{\log }_{5}}6\cdot {{\log }_{6}}7\cdot {{\log }_{7}}8$

$=\left( {{\log }_{2}}3\cdot {{\log }_{3}}4 \right)\cdot {{\log }_{4}}5\cdot {{\log }_{5}}6\cdot {{\log }_{6}}7\cdot {{\log }_{7}}8$

$={{\log }_{2}}4\cdot {{\log }_{4}}5\cdot {{\log }_{5}}6\cdot {{\log }_{6}}7\cdot {{\log }_{7}}8$

$=\left( {{\log }_{2}}4\cdot {{\log }_{4}}5 \right)\cdot {{\log }_{5}}6\cdot {{\log }_{6}}7\cdot {{\log }_{7}}8$

$={{\log }_{2}}5\cdot {{\log }_{5}}6\cdot {{\log }_{6}}7\cdot {{\log }_{7}}8$

$=\left( {{\log }_{2}}5\cdot {{\log }_{5}}6 \right)\cdot {{\log }_{6}}7\cdot {{\log }_{7}}8$

$={{\log }_{2}}6\cdot {{\log }_{6}}7\cdot {{\log }_{7}}8$

$=\left( {{\log }_{2}}6\cdot {{\log }_{6}}7 \right)\cdot {{\log }_{7}}8$

$={{\log }_{2}}7\cdot {{\log }_{7}}8$

$={{\log }_{2}}8={{\log }_{2}}{{2}^{3}}=3$