$$\frac{1}{log_{2}100}-\frac{1}{log_{4}100}+\frac{1}{log_{5}100}-\frac{1}{log_{10}100}+\frac{1}{log_{20}100}-\frac{1}{log_{25}100}+\frac{1}{log_{50}100}$$=?

We know that $$\dfrac{1}{log_{a}{b}}$$ = $$\dfrac{log_{x}{a}}{log_{x}{b}}$$

Therefore, we can say that $$\dfrac{1}{log_{2}{100}}$$ = $$\dfrac{log_{10}{2}}{log_{10}{100}}$$

$$\Rightarrow$$ $$\frac{1}{log_{2}100}-\frac{1}{log_{4}100}+\frac{1}{log_{5}100}-\frac{1}{log_{10}100}+\frac{1}{log_{20}100}-\frac{1}{log_{25}100}+\frac{1}{log_{50}100}$$

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

We know that $$log_{10}{100}=2$$

$$\Rightarrow$$ $$\dfrac{1}{2}*[log_{10}{2}-log_{10}{4}+log_{10}{5}-log_{10}{10}+log_{10}{20}-log_{10}{25}+log_{10}{50}]$$

$$\Rightarrow$$ $$\dfrac{1}{2}*[log_{10}{\dfrac{2*5*20*50}{4*10*25}}]$$

$$\Rightarrow$$ $$\dfrac{1}{2}*[log_{10}10]$$

$$\Rightarrow$$ $$\dfrac{1}{2}$$