Find $\left\lfloor\prod_{j=3}^{2020}\log_{j-1}j\right\rfloor$.

Question

$\log_2 3\cdot\log_3 4\cdot\log_4 5\cdots\log_{2019} 2020=x$

Find the largest natural number which is less than $x$.

Answer 1

Hint:

$$\log_{a} b = \frac{\log b}{\log a}$$

Answer 2

Your product is $\prod_{j=3}^{2020}\frac{\ln j}{\ln (j-1)}$, which telescopes to $\frac{\ln(2020)}{\ln 2}$. Its integer part is $10$.