(Log5 2+ log2 5+2)*(log5 2-lg2)*log2 5-log5 2

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To simplify this expression, we will first expand and combine the log terms:

Step 1: Expand the expression.

= (log5 2 + log2 5 + 2) (log5 2 - log2) log2 5 - log5 2
= (log5 2 + log2 5 + 2) (log5 2 - log2) log2 5 - log5 2

Step 2: Use the properties of logarithms to combine the terms.

= (log5 2 + log2 5 + 2) (log5 2 - log2) log2 5 - log5 2
= (log5 2 + log2 5 + 2) (log2 5 - log5) log2 5 - log5 2
= (log5 2 + log2 5 + 2) (-log5 2) log2 5 - log5 2
= -((log5 2 + log2 5 + 2) log5 2 log2 5) - log5 2

Step 3: Use the properties of logarithms to simplify further.

= -((log5 2 + log2 5 + 2) log5 2 log2 5) - log5 2
= -((1 + 1) 1 1) - log5 2
= -2 - log5 2

Therefore, the final simplified expression is -2 - log5 2.