$\dfrac{1}{\log _{2} 100}-\dfrac{1}{\log _{4} 100}+\dfrac{1}{\log _{5} 100}-\dfrac{1}{\log _{10} 100}+\dfrac{1}{\log _{20} 100}-\dfrac{1}{\log _{25} 100}+\dfrac{1}{\log _{50} 100}= \ ?$

Use logarithm property:

$\dfrac{1}{\log_b a} = \log_a b$

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Applying this property to each term while keeping the correct signs:

$\Rightarrow \frac{1}{\log_2 100} - \frac{1}{\log_4 100} + \frac{1}{\log_5 100} - \frac{1}{\log_{10} 100} + \frac{1}{\log_{20} 100} - \frac{1}{\log_{25} 100} + \frac{1}{\log_{50} 100}$

$\Rightarrow \log_{100} 2 - \log_{100} 4 + \log_{100} 5 - \log_{100} 10 + \log_{100} 20 - \log_{100} 25 + \log_{100} 50$

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Group the positive and negative terms separately:

Positive terms: $\log_{100} 2 + \log_{100} 5 + \log_{100} 20 + \log_{100} 50$

Negative terms: $\log_{100} 4 + \log_{100} 10 + \log_{100} 25$

Using $\log_a x + \log_a y = \log_a(xy)$:

Positive: $\log_{100}(2 \times 5 \times 20 \times 50) = \log_{100}(10,000)$

Negative: $\log_{100}(4 \times 10 \times 25) = \log_{100}(1,000)$

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Combine using the subtraction property $\log_a x - \log_a y = \log_a\left(\frac{x}{y}\right)$:

$\log_{100}(10,000) - \log_{100}(1,000) = \log_{100}\left(\frac{10,000}{1,000}\right) = \log_{100}(10)$

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Since $100 = 10^2$, we can evaluate:

$\log_{100} 10 = \log_{10^2} 10 = \frac{\log 10}{\log 10^2} = \frac{1}{2}$

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The answer is $\frac{1}{2}$