Решебник по алгебре 11 класс Никольский Параграф 10. Равносильность уравнений на множествах Задание 52

\[\boxed{\mathbf{52.}}\]

\[\textbf{а)}\ \left( x - \log_{3}75 \right)\left( x - \log_{2}22 \right) =\]

\[= 0\]

\[x \in \lbrack 3;4\rbrack.\]

\[1)\ x - \log_{3}75 = 0\]

\[x = \log_{3}75\]

\[\log_{3}27 < \log_{3}75 < \log_{3}81\]

\[3 < \log_{3}75 < 4.\]

\[2)\ x - \log_{2}22 = 0\]

\[x = \log_{2}22\]

\[\log_{2}16 < \log_{2}22\]

\[4 < \log_{2}22.\]

\[Ответ:x = \log_{3}75.\]

\[\textbf{б)}\ \left( x - \log_{2}17 \right)\left( x - \log_{2}71 \right) =\]

\[= 0\]

\[x \in \lbrack 4;5\rbrack.\]

\[1)\ x - \log_{2}17 = 0\]

\[x = \log_{2}17\]

\[\log_{2}16 < \log_{2}17 < \log_{2}32\]

\[4 < \log_{2}17 < 5.\]

\[2)\ x - \log_{2}71 = 0\]

\[x = \log_{2}71\]

\[\log_{2}64 < \log_{2}71\]

\[6 < \log_{2}71.\]

\[Ответ:x = \log_{2}17.\]