TITLE: Simplify $\log_23\ \log_34\ \log_4 5\ \log_5 6\ \log_6 7\ \log_7 8$
QUESTION [5 upvotes]: How do I evaluate the product:
$$\log_23\ \log_34\ \log_4 5\ \log_5 6\ \log_6 7\ \log_7 8$$
I know that $$\log_ba=\frac{\log\ a}{\log\ b}$$
How can I apply it?
Thanks!

REPLY [2 votes]: Hint:  $log_23=\frac{log3}{log2}, log_34=\frac{log4}{log3}$ and so on, find anything fishy?