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If k=3^81 and k^k=3^m, what is the value of m?

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If k=3^81 and k^k=3^m, what is the value of m?  [#permalink]

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New post 11 Jul 2018, 05:36
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Question Stats:

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If \(k=3^{81}\) and \(k^k=3^m\), what is the value of m?

A. \(85\)
B. \(24^{381}\)
C. \(3^{54}\)
D. \(3^{85}\)
E. \(3^{72}\)
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Re: If k=3^81 and k^k=3^m, what is the value of m?  [#permalink]

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New post 11 Jul 2018, 06:06
Karthik200 wrote:
If \(k=3^{81}\) and \(k^k=3^m\), what is the value of m?

A. \(85\)
B. \(24^{381}\)
C. \(3^{54}\)
D. \(3^{85}\)
E. \(3^{72}\)



I think answer should be D. kindly verify the official answer.
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Re: If k=3^81 and k^k=3^m, what is the value of m?  [#permalink]

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New post 11 Jul 2018, 06:18
Karthik200 wrote:
If \(k=3^{81}\) and \(k^k=3^m\), what is the value of m?

A. \(85\)
B. \(24^{381}\)
C. \(3^{54}\)
D. \(3^{85}\)
E. \(3^{72}\)


Looks like you got the OA wrong here. My answer comes out as D.
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Re: If k=3^81 and k^k=3^m, what is the value of m?  [#permalink]

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New post 11 Jul 2018, 06:26
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Karthik200 wrote:
If \(k=3^{81}\) and \(k^k=3^m\), what is the value of m?

A. \(85\)
B. \(24^{381}\)
C. \(3^{54}\)
D. \(3^{85}\)
E. \(3^{72}\)


Given \(k=3^{81}\)

Now, \(k^k=({3^{81}})^{3^{81}}\)=\({3^{{81}*3^{81}}}\)=\({3(^{{3^4}*3^{81}})}\)=\(3^{3^{85}}\)

Given \(k^k=3^m\)
So, \(3^m\)=\(3^{3^{85}}\)
Or, \(m=3^{85}\)

Ans. (D)

P.S:- Property used in solving this question:-\(a^m*a^n=a^{m+n} & ({a^m})^n={a}^{mn}\)
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Re: If k=3^81 and k^k=3^m, what is the value of m?  [#permalink]

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New post 11 Jul 2018, 22:23
k=3^81
k^k = 3^m
taking log in both equations, we get
log k = 81 log 3 ------ (1)
and
k*log k = m log 3 ------ (2)
Dividing (2) by (1)
k = m/81
=> m = k*81
or
m = 3^81*81
=3^85

D is the right answer
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Re: If k=3^81 and k^k=3^m, what is the value of m?  [#permalink]

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New post 14 Jul 2018, 08:36
\(81 = 3^4\)

Using the property: \(a^n * a^m = a^{n+m}\)

\(3^4 * 3^{81} = 3^{4+81}\) or \(3^{85}\)

Option D is the correct answer
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Re: If k=3^81 and k^k=3^m, what is the value of m?  [#permalink]

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New post 17 Jul 2018, 16:25
Karthik200 wrote:
If \(k=3^{81}\) and \(k^k=3^m\), what is the value of m?

A. \(85\)
B. \(24^{381}\)
C. \(3^{54}\)
D. \(3^{85}\)
E. \(3^{72}\)


Since k = 3^81, k^k = (3^81)^(3^81) = (3^(3^4))^(3^81) = 3^[(3^4)*(3^81)] = 3^(3^85). So, m must be 3^85.

Answer: D
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Re: If k=3^81 and k^k=3^m, what is the value of m? &nbs [#permalink] 17 Jul 2018, 16:25
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