First slide

Types of real valued functions XI

Question

If $(21 .4)^{a} = (0 .00214)^{b} = 100,$ then the value of $\frac{1}{a} - \frac{1}{b}$ is

Moderate

A

0
B

1
C

2
D

4

Solution

$\begin{array}{l} (21 .4)^{a} = 100 \\ \Rightarrow alog (21 .4) = 2 \\ ∴ \log (21 .4) = 2 / a \end{array}$
$from (0 .00214)^{b} = 100 we get$
$\begin{array}{l} or b (\log 0 .00214) = 2 \\ or blog (21 .4 \times 10^{- 4}) = 2 \\ b = \frac{2}{\log 21 .4 - 4} = \frac{2}{\frac{2}{a} - 4} = \frac{1}{\frac{1}{a} - 2} = \frac{a}{1 - 2 a} \\ \Rightarrow \frac{1}{b} = \frac{1 - 2 a}{a} \\ \Rightarrow \frac{1}{a} - \frac{1}{b} = 2 \end{array}$

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