If {log ₓ}a,{a^{x/2}} and {log ₓ}x are in G.P. , then x =

English

Gujarati

Hindi

8. Sequences and Series

easy

If ${\log _x}a,\;{a^{x/2}}$ and ${\log _b}x$ are in $G.P.$, then $x = $

A

$ - \log ({\log _b}a)$

B

$ - {\log _a}({\log _a}b)$

C

${\log _a}({\log _e}a) - {\log _a}({\log _e}b)$

D

${\log _a}({\log _e}b) - {\log _a}({\log _e}a)$

Solution

(c) Obviously ${({a^{x/2}})^2} = {\log _x}a\;.\;{\log _b}x = {\log _b}a$

$ \Rightarrow $${a^x} = {\log _b}a$

$ \Rightarrow $$x = {\log _a}({\log _b}a)$

$ \Rightarrow $$x = {\log _a}\left( {\frac{{{{\log }_e}a}}{{{{\log }_e}b}}} \right) = {\log _a}({\log _e}a) – {\log _a}({\log _e}b)$.

Standard 11

Mathematics

Share

Similar Questions

The solution of the equation $1 + a + {a^2} + {a^3} + ……. + {a^x}$ $ = (1 + a)(1 + {a^2})(1 + {a^4})$ is given by $x$ is equal to

easy

A person has $2$ parents, $4$ grandparents, $8$ great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.

easy

Find four numbers forming a geometric progression in which the third term is greater than the first term by $9,$ and the second term is greater than the $4^{\text {th }}$ by $18 .$

medium

If the sum and product of four positive consecutive terms of a $G.P.$, are $126$ and $1296$, respectively, then the sum of common ratios of all such $GPs$ is $………$.

hard

(JEE MAIN-2023)

The sum of first three terms of a $G.P.$ is $\frac{13}{12}$ and their product is $-1$ Find the common ratio and the terms.

medium