Question

Вычислить: ((3+㏒₁₂27) / (3-㏒₁₂27)) · ㏒₆16

Answer 1

$\frac{3+log_{12}27}{3-log_{12}27}\cdot log_616=\\\\=\Big [\; log_{12}27=log_{12}3^3=3\cdot log_{12}3=3\cdot \frac{log_23}{log_212}=3\cdot \frac{log_23}{log_2(2^2\cdot 3)}=\\\\=3\cdot \frac{log_23}{log_22^2+log_23}=3\cdot \frac{log_23}{2+log_23}\; ;\\\\log_616=\frac{log_216}{log_26}=\frac{log_22^4}{log_2(2\cdot 3)}=\frac{4\cdot log_22}{log_22+log_23}=\frac{4}{1+log_23}\; \Big ]=$

$=\frac{3+3\cdot \frac{log_23}{2+log_23}}{3-3\cdot \frac{log_23}{2+log_23}}\cdot \frac{4}{1+log_23}=\frac{3\cdot (2+log_23+log_23)}{3\cdot (2+log_23-log_23)}\cdot \frac{4}{1+log_23}=\\\\=\frac{(2+2log_23)\cdot 4}{2\cdot (1+log_23)}=\frac{2\cdot (1+log_23)\cdot 4}{2\cdot (1+log_23)}=4$