1.2.1 \( \frac{\log 5+\log 2}{\log 20+\log 5} \)

To simplify the expression \( \frac{\log 5+\log 2}{\log 20+\log 5} \), we can use the property of logarithms that states \( \log a + \log b = \log(ab) \). First, let's simplify the numerator: \[ \log 5 + \log 2 = \log(5 \cdot 2) = \log 10 \] Now for the denominator: \[ \log 20 + \log 5 = \log(20 \cdot 5) = \log 100 \] Putting these simplified forms back into our expression, we have: \[ \frac{\log 10}{\log 100} \] Next, we can further simplify it. We know that: \[ \log 10 = 1 \quad \text{(since } 10^1 = 10\text{)} \] \[ \log 100 = 2 \quad \text{(since } 10^2 = 100\text{)} \] Thus, we can rewrite the expression as follows: \[ \frac{\log 10}{\log 100} = \frac{1}{2} \] So, the final answer is: \[ \frac{1}{2} \]