Question

LOG

Answer 1

Let N =
\[\huge \frac{ \log_3^{135} }{ \log_{15}^{3} } - \frac{ \log_3^{5} }{ \log_{405}^{3} }\]

then N is

(A) a natural number
(B)a prime number
(C)a rational number
(D)an integer

Answer 2

@ganeshie8

Answer 3

Use change of base.

Answer 4

Yes wait

Answer 5

\[\huge \log_3^{135} * \log_3^{15} - \log_3^{5}*\log_3^{405}\]

Answer 6

Good.

Answer 7

My internet is not working properly sorry for slow reply

Answer 8

How to simplify it further

Answer 9

But the asks about N (the expression)

Answer 10

yes

Answer 11

\[(3 + \log 5)(1 + \log 5) =3 + 4\log 5 +(\log 5)^2\]
\[\log(405) = \log(81) + \log(5)=4 + \log(5)\]\[\log5(4 + \log5) = 4\log 5 + (\log 5)^2\]

Answer 12

\[\therefore 3 + 4\log 5 + (\log 5)^2 - 4\log 5 - (\log 5)^2 = 3\]

Answer 13

There was a silly mistake earlier :P

Answer 14

Thanks