Which of the following, when simplified, reduces to unity? `(log)_(10)5dot(log)_(10)20+((log)_(10)2)^2` . (c) `-(log)_5(log)_3sqrt(5sqrt(9))` `1/6(log)_((sqrt(3))/2)((64)/(27))`

Options

A. ` log_(10) 5 * log_(10) 20+(log_(10) 2)^(2)`
B. `(2 log 2+log 3)/(log 48 - log 4)`
C. ` - log_(5) log_(3) sqrtroot(5)9`
D. `1/6 log_(sqrt3//2)(64/27)`

Explanation

To solve the question, we need to simplify the given expressions and determine which of them reduces to unity (1). Let's analyze each option step by step.

### Given Expression:
1. **Option 1:** \( \log_{10} 5 \cdot \log_{10} 20 + (\log_{10} 2)^2 \)

### Step-by-Step Solution:

**Step 1: Rewrite the logarithms using properties.**
- We can use the property of logarithms that states \( \log_{a} b + \log_{a} c = \log_{a} (b \cdot c) \) and \( \log_{a} b - \log_{a} c = \log_{a} \left( \frac{b}{c} \right) \).

**Step 2: Simplify \( \log_{10} 20 \).**
- We know that \( 20 = 10 \cdot 2 \), so:
\[ \log_{10} 20 = \log_{10} (10 \cdot 2) = \log_{10} 10 + \log_{10} 2 = 1 + \log_{10} 2 \]

**Step 3: Substitute back into the expression.**
- Now substitute \( \log_{10} 20 \) back into the expression:
\[ \log_{10} 5 \cdot (1 + \log_{10} 2) + (\log_{10} 2)^2 \]

**Step 4: Distribute \( \log_{10} 5 \).**
- This gives us:
\[ \log_{10} 5 + \log_{10} 5 \cdot \log_{10} 2 + (\log_{10} 2)^2 \]

**Step 5: Recognize the quadratic form.**
- Notice that we can rearrange this as:
\[ \log_{10} 5 + \log_{10} 2 \cdot \log_{10} 5 + (\log_{10} 2)^2 = \log_{10} 5 + (\log_{10} 2)(\log_{10} 5 + \log_{10} 2) \]
- This is a quadratic in terms of \( \log_{10} 2 \):
\[ = \log_{10} 5 + (\log_{10} 2)^2 + \log_{10} 5 \cdot \log_{10} 2 \]

**Step 6: Use the identity \( a^2 + ab + b^2 = (a + b)^2 \).**
- Let \( a = \log_{10} 2 \) and \( b = \log_{10} 5 \):
\[ = (\log_{10} 2 + \log_{10} 5)^2 \]

**Step 7: Simplify further.**
- Since \( \log_{10} 2 + \log_{10} 5 = \log_{10} (2 \cdot 5) = \log_{10} 10 = 1 \):
\[ = 1^2 = 1 \]

### Conclusion:
Thus, the first option simplifies to unity.