$$\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}-\

Question

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

Accepted Answer

1 2 \frac{1}{2} 2 1 ​. We know that 1 l o g a b \dfrac{1}{log_{a}{b}} l o g a ​ b 1 ​ = l o g x a l o g x b \dfrac{log_{x}{a}}{log_{x}{b}} l o g x ​ b l o g x ​ a ​ Therefore, we can say that 1 l o g 2 100 \dfrac{1}{log_{2}{100}} l o g 2 ​ 100 1 ​ = l o g 10 2 l o g 10 100 \dfrac{log_{10}{2}}{log_{10}{100}} l o g 10 ​ 100

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

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