Question

If $(21.4{)}^a=(0.00214{)}^b=100$, then the value of $\frac{1}{a}-\frac{1}{b}$ is

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is C $2$

$(21.4{)}^a=(0.00214{)}^b=100$
$\Rightarrow a{\log }_{10}\left(21.4\right)=2,b{\log }_{10}\left(0.00214\right)=2$
$\Rightarrow \frac{1}{a}=\frac{1}{2}{\log }_{10}\left(21.4\right),\frac{1}{b}=\frac{1}{2}{\log }_{10}\left(0.00214\right)\therefore \frac{1}{a}-\frac{1}{b}=\frac{1}{2}({\log }_{10}\left(21.4\right)-{\log }_{10}\left(0.00214\right))\Rightarrow \frac{1}{a}-\frac{1}{b}=\frac{1}{2}{\log }_{10}\left(\frac{21.4}{0.00214}\right)$
$\Rightarrow \frac{1}{a}-\frac{1}{b}=\frac{1}{2}{\log }_{10}\left(10000\right)$
$\therefore \frac{1}{a}-\frac{1}{b}=\frac{1}{2}\times 4=2$