I'm studying logarithms and I encountered this equation [log_9(k+1)]^2+log_9(k+1)+(k+1)>3 I tried a lot but I still couldn't solve it! I know this may be easy for most of you but please could you help me?

Haiphongum

Haiphongum

Answered question

2022-09-26

Logarithmic equation
I'm studying logarithms and I encountered this equation:
[ log 9 ( k + 1 ) ] 2 + log 9 ( k + 1 ) + ( k + 1 ) > 3
I tried a lot but I still couldn't solve it! I know this may be easy for most of you but please could you help me?
Thanks!

Answer & Explanation

vyhlodatis

vyhlodatis

Beginner2022-09-27Added 14 answers

With k N you have log 9 ( k + 1 ) > 0. Therefore it is obvious that all k > 1 satisfy the in equality (because k + 1 3). For k = 0 it is obviously wrong, so the only case to check is k = 1, and here the LHS is 2.41
Summary: The integer solutions are all k > 1, i.e. k = 2 , 3 ,
Kolby Castillo

Kolby Castillo

Beginner2022-09-28Added 1 answers

I didn't see if the solution must be integer or real. In every case, pose t = log 9 ( k + 1 ), then solve t 2 + t + ( k 2 ) > 0, from which
t ± = 1 ± 1 4 ( k 2 ) 2 .
So in order to have real solutions we must impose k 2 + 1 4
Then we have
[ log 9 ( k + 1 ) < 1 1 4 ( k 2 ) 2 ] [ log 9 ( k + 1 ) > 1 + 1 4 ( k 2 ) 2 ] .

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