Question

Find the value of${\log }_{2}3·{\log }_{3}4·{\log }_{4}5·{\log }_{5}6·{\log }_{6}7·{\log }_{7}8$.

Short answer

The obtained result is 3.

Step-by-step solution

Step 1. Given information.

Consider the given expression.

${\log }_{2}3·{\log }_{3}4·{\log }_{4}5·{\log }_{5}6·{\log }_{6}7·{\log }_{7}8$

Step 2. Calculation.

Apply the base change formula to simplify the expression.

$\begin{array}{rcl}{\log }_{2}3·{\log }_{3}4·{\log }_{4}5·{\log }_{5}6·{\log }_{6}7·{\log }_{7}8& =& \frac{\log 3}{\log 2}·\frac{\log 4}{\log 3}·\frac{\log 5}{\log 4}·\frac{\log 6}{\log 5}·\frac{\log 7}{\log 6}·\frac{\log 8}{\log 7}\\ & =& \frac{1}{\log 2}·\frac{1}{1}·\frac{1}{1}·\frac{1}{1}·\frac{1}{1}·\frac{\log 2^3}{1}\\ & =& \frac{3\log 2}{\log 2}\\ & =& 3\end{array}$