Question

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

• A. 0
• B. 1
• C. 2
• D. 4

Answer 1

The correct option is C 2

Given that
$(21.4{)}^a=(0.00214{)}^b=100$
$(21.4{)}^a=100,(0.00214{)}^b=100$
Applying $\log$ on both the sides, we get
$a={\log }_{21.4}100,b={\log }_{0.00214}100$
$\therefore \frac{1}{a}-\frac{1}{b}=\frac{1}{{\log }_{21.4}100}-\frac{1}{{\log }_{0.00214}100},$ $\because$ $[{\log }_{a}b=\frac{1}{{\log }_{b}a}]$
$\Rightarrow \frac{1}{a}-\frac{1}{b}={\log }_{100}21.4-{\log }_{100}0.00214$
$\therefore \frac{1}{a}-\frac{1}{b}={\log }_{100}\frac{21.4}{0.00214}={\log }_{100}{100}^2=2{\log }_{100}100=2,$
$\because$ $[\log a-\log b=\log \frac{a}{b}]$