Value of x satisfying {log ₐ}x + {log _{√ a }}x + {log _{3√ a }}x + .........{log _{a√ a }}x

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8. Sequences and Series

easy

The value of $x$ satisfying ${\log _a}x + {\log _{\sqrt a }}x + {\log _{3\sqrt a }}x + .........{\log _{a\sqrt a }}x = \frac{{a + 1}}{2}$ will be

A

$x = a$

B

$x = {a^a}$

C

$x = {a^{ - 1/a}}$

D

$x = {a^{1/a}}$

Solution

(d) $ {\log _a}x + 2{\log _a}x + ……. + a{\log _a}x = \frac{{a + 1}}{2}$

$ \Rightarrow $ ${\log _a}x(1 + 2 + …….. + a) = \frac{{a + 1}}{2}$

$ \Rightarrow $ ${\log _a}x.\frac{{a(a + 1)}}{2} = \frac{{a + 1}}{2}$

$ \Rightarrow $ $x=a^{1/a}$.

Standard 11

Mathematics

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