Question

Calculati:

㏒₂3 * ㏒₃4 * ㏒₄5 * ... * ㏒₇8

Answer 1

$\displaystyle log_23 \cdot log_34 \cdot log_45 \cdot ... \cdot log_78=log_23 \cdot log_34 \cdot log_45 \cdot log_56 \cdot log_67 \cdot log_78= \\ \\ = \frac{lg3}{lg2} \cdot \frac{lg4}{lg3} \cdot \frac{lg5}{lg4}\cdot \frac{lg6}{lg5} \cdot \frac{lg7}{lg6} \cdot \frac{lg8}{lg7} = \frac{1}{lg2} \cdot lg8= \frac{lg8}{lg2} = \\ \\ =log_28=log_22^3=3log_22=\boxed{3}$