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Continuity of a Function

If a , b , ...

Question

If $a, b, c \in R^{*} - {1}$ and the numbers ${log}_{a} 100, 2 {log}_{b} 100, 2 {log}_{c} 5 + {log}_{c} 4$ are in H.P. then

2 b = a + c

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b = a c

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b (a + c) = 2 a c

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none

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Solution

The correct option is B $b = a c$
$l o g_{a} 100, 2 l o g_{b} 100, 2 l o g_{c} 5 + l o g_{c} 4$ in HP

$\Rightarrow \frac{l o g 100}{l o g a}, \frac{2 l n 100}{l n b}, \frac{2 l n 5}{l n c} + \frac{l n 4}{l n c}$ in HP

$\Rightarrow \frac{l n 100}{l n a}, \frac{2 l n 100}{l n b}, \frac{l n 25}{l n c} + \frac{l n 4}{l n c}$ in HP

$\Rightarrow \frac{l n 100}{l n a}, \frac{2 l n 100}{l n b}, \frac{l n 100}{l n c}$ in HP

$\Rightarrow \frac{l n a}{l n 100}, \frac{l n b}{2 l n 100}, \frac{l n c}{l n 100}$ in AP

$\Rightarrow \frac{2 l n b}{2 l n 100} = \frac{l n a}{l n 100} + \frac{l n c}{l n 100}$

$\Rightarrow l n b = l n a + l n c$

$\Rightarrow l n b = l n (a c)$

$\Rightarrow b = a c$

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