Consider \(a = {\log _2}3 \times {\log _3}4 \times {\log _4}5 \times  \ldots  \times {\log _{31}}32\). Given that \(a \in \mathbb{Z}\), find the value of a.

\(\frac{{\log 3}}{{\log 2}} \times \frac{{\log 4}}{{\log 3}} \times  \ldots \times \frac{{\log 32}}{{\log 31}}\)     M1A1

\( = \frac{{\log 32}}{{\log 2}}\)     A1

\( = \frac{{5\log 2}}{{\log 2}}\)     (M1)

\( = 5\)     A1

hence \(a = 5\)

Note:     Accept the above if done in a specific base eg \({\log _2}x\).

[5 marks]