Cho n là số nguyên dương và a>0, a khác 1. Tìm n sao cho

* Hướng dẫn giải

Đáp án B

Ta có: $\begin{array}{l}{\log }_{a}2019+{\log }_{\sqrt{a}}2019+\mathrm{...}+{\log }_{\sqrt[n]{a}}2019\end{array}$

$\begin{array}{l}={\log }_{a}2019+2{\log }_{2019}+\mathrm{...}+n{\log }_{a}2019\end{array}$

$\begin{array}{l}={\log }_{a}2019\left(1+2+\mathrm{...}+n\right)=\frac{n}{2}\left(n+1\right){\log }_{a}2019\end{array}$

$\begin{array}{l}=2033136{\log }_{a}2019\Rightarrow \frac{n}{2}\left(n+1\right)=2033136\end{array}$

$\iff n^2+n-4066272=0\iff \left[\begin{array}{l}n=2016\\ n=-2017\end{array}\right.\Rightarrow n=2016.$