The value of x satisfying log a x-log -a x-log 3 -a x- - log a -a x-a-1-2 will be

The correct option is D log_ax + 2log_ax + .... + alog_ax = a+1/2 ⇒log_ax (1+2+.... + a) =a+1/2 ⇒log_ax.a(a+1)/2 =a+1/2⇒ x = a^1/a. ...

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Byju's Answer

Standard XII

Mathematics

Special Integrals - 2

The value of ...

Question

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

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Solution

The correct option is D
${log}_{a} x + 2 {log}_{a} x + . . . . + a {log}_{a} x = \frac{a + 1}{2}$
$\Rightarrow {log}_{a} x (1 + 2 + . . . . + a) = \frac{a + 1}{2}$
$\Rightarrow {log}_{a} x . \frac{a (a + 1)}{2} = \frac{a + 1}{2} \Rightarrow x = a^{\frac{1}{a}} .$

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Similar questions

Find the value of x satisfying the equation $l o g_{a} x + l o g_{\sqrt{a}} x + l o g_{\sqrt[3]{a}} x + . . . . . . . . . . l o g_{\sqrt[a]{a}} x$ = $\frac{a + 1}{2}$ .

Q. Solve the system of equations:

l o g_{a} x l o g_{a} (x y z) = 48 l o g_{a} y l o g_{a} (x y z) = 12, a > 0, a \neq 1 l o g_{a} z l o g_{a} (x y z) = 84

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The value of x satisfying logax+log√ax+log3√ax+....loga√ax=a+12 will be

The correct option is D logax+2logax+....+alogax=a+12 ⇒logax(1+2+....+a)=a+12 ⇒logax.a(a+1)2=a+12⇒x=a1a.

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

The correct option is D
${log}_{a} x + 2 {log}_{a} x + . . . . + a {log}_{a} x = \frac{a + 1}{2}$
$\Rightarrow {log}_{a} x (1 + 2 + . . . . + a) = \frac{a + 1}{2}$
$\Rightarrow {log}_{a} x . \frac{a (a + 1)}{2} = \frac{a + 1}{2} \Rightarrow x = a^{\frac{1}{a}} .$