I have the following inquality

log(a_n+c)/log(a_n)<=(n+1)/n

where c>0 is a fixed constant and (a_n)_n is a positive real sequence.

I think I can't solve it explicetly.
But I would like to prove that if there exist K>0 and k>0 such that

a_n >= K e^(k n) for any n positive integer

then the inequality is verified for any n big enough.

Do you think it's true? Do you know how to prove it?