Find the value of the expressions `(log2)^3+log8.log(5)+(log5)^3` .

Answer

To find the value of the expression \( (\log 2)^3 + \log 8 \cdot \log 5 + (\log 5)^3 \), we can follow these steps:

### Step 1: Rewrite \(\log 8\)
We know that \( 8 = 2^3 \). Therefore, we can rewrite \(\log 8\) using the power rule of logarithms:
\[ \log 8 = \log(2^3) = 3 \log 2 \]

### Step 2: Substitute \(\log 8\) into the expression
Now, substitute \(\log 8\) back into the original expression:
\[ (\log 2)^3 + \log 8 \cdot \log 5 + (\log 5)^3 = (\log 2)^3 + (3 \log 2) \cdot \log 5 + (\log 5)^3 \]

### Step 3: Recognize the form of the expression
The expression now resembles the expansion of \( (a + b)^3 \), where \( a = \log 2 \) and \( b = \log 5 \). The expansion of \( (a + b)^3 \) is given by:
\[ a^3 + b^3 + 3ab(a + b) \]
Thus, we can write:
\[ (\log 2)^3 + (\log 5)^3 + 3 \log 2 \cdot \log 5 \cdot (\log 2 + \log 5) \]

### Step 4: Simplify using logarithmic properties
Using the property of logarithms that states \( \log a + \log b = \log(ab) \), we can simplify \( \log 2 + \log 5 \):
\[ \log 2 + \log 5 = \log(2 \cdot 5) = \log 10 \]
Since \( \log 10 = 1 \), we can substitute this back into our expression:
\[ (\log 2)^3 + (\log 5)^3 + 3 \log 2 \cdot \log 5 \cdot 1 \]

### Step 5: Final expression
Now, we have:
\[ (\log 2)^3 + (\log 5)^3 + 3 \log 2 \cdot \log 5 \]
This is equal to:
\[ (\log 2 + \log 5)^3 = (\log 10)^3 = 1^3 = 1 \]

### Conclusion
Thus, the value of the expression \( (\log 2)^3 + \log 8 \cdot \log 5 + (\log 5)^3 \) is:
\[ \boxed{1} \]