Question

Find the value of${\log }_{2}3·{\log }_{3}4·\cdots ·{\log }_n\left(n+1\right)·{\log }_{n+1}2$

Short answer

The obtained result is 1.

Step-by-step solution

Step 1. Given information.

Consider the given expression.

${\log }_{2}3·{\log }_{3}4·\cdots ·{\log }_n\left(n+1\right)·{\log }_{n+1}2$

Step 2. Calculation.

Apply the base change formula to simplify the expression.

$\begin{array}{rcl}{\log }_{2}3·{\log }_{3}4·\cdots ·{\log }_n\left(n+1\right)·{\log }_{n+1}2& =& \frac{\log 3}{\log 2}·\frac{\log 4}{\log 3}·\cdots ·\frac{\log \left(n+1\right)}{\log n}·\frac{\log 2}{\log \left(n+1\right)}\\ & =& \frac{1}{\log 2}·\frac{1}{1}·\cdots ·\frac{1}{1}·\frac{\log 2}{1}\\ & =& \frac{\log 2}{\log 2}\\ & =& 1\end{array}$