Simplification of logarithmic alternating sum

Using $\frac{1}{\log_a b} = \log_b a$, the given expression becomes: $\log_{100} 2 - \log_{100} 4 + \log_{100} 5 - \log_{100} 10 + \log_{100} 20 - \log_{100} 25 + \log_{100} 50$

Combine using logarithmic properties: $= \log_{100} \left( \frac{2 \times 5 \times 20 \times 50}{4 \times 10 \times 25} \right)$

Numerator $= 2 \times 5 \times 20 \times 50 = 10000$. Denominator $= 4 \times 10 \times 25 = 1000$.

$\frac{\text{Numerator}}{\text{Denominator}} = \frac{10000}{1000} = 10$

So expression $= \log_{100} 10 = \frac{1}{2}$.