Логаритамске неједначине 1

Текст задатака објашњених у видео лекцији.

Решити логаритамску неједначину.

пр.1)   ${\log _2}x > 0$

пр.2)   ${\log _{\frac{1}{{64}}}}x >  - \frac{1}{2}$

пр.3)   ${\log _x}125 < 3$

Пр.1

\[\begin{gathered}
{\log _2}x > 0 \hfill \\
\left\{ \begin{gathered}
x > 0 \hfill \\
{\log _2}x > {\log _2}1 \hfill \\
\end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered}
x > 0 \hfill \\
x > 1 \hfill \\
\end{gathered} \right. \hfill \\
x \in \left( {1, + \infty } \right) \hfill \\
\end{gathered} \]

Пр.2

\[\begin{gathered}
{\log _{\frac{1}{{64}}}}x > - \frac{1}{2} \hfill \\
\left\{ \begin{gathered}
x > 0 \hfill \\
{\log _{\frac{1}{{64}}}}x > - \frac{1}{2}{\log _{\frac{1}{{64}}}}\frac{1}{{64}} \hfill \\
\end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered}
x > 0 \hfill \\
{\log _{\frac{1}{{64}}}}x > {\log _{\frac{1}{{64}}}}{\left( {\frac{1}{{64}}} \right)^{ - \frac{1}{2}}} \hfill \\
\end{gathered} \right. \Leftrightarrow \hfill \\
0 < a < 1 \hfill \\
\Leftrightarrow \left\{ \begin{gathered}
x > 0 \hfill \\
x < {\left( {\frac{1}{{64}}} \right)^{ - \frac{1}{2}}} \hfill \\
\end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered}
x > 0 \hfill \\
x < \sqrt {64} \hfill \\
\end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered}
x > 0 \hfill \\
x < 8 \hfill \\
\end{gathered} \right. \hfill \\
x \in \left( {0;8} \right) \hfill \\
\end{gathered} \]

Пр.3

\[\begin{gathered}
{\log _x}125 < 3 \hfill \\
\hfill \\
\left\{ \begin{gathered}
x > 0 \hfill \\
x \ne 1 \hfill \\
{\log _x}125 < 3{\log _x}x \hfill \\
\end{gathered} \right.\left\{ \begin{gathered}
x > 0 \hfill \\
x \ne 1 \hfill \\
{\log _x}125 < {\log _x}{x^3} \hfill \\
\end{gathered} \right. \hfill \\
\end{gathered} \]

Размислимо o два случаја

\[\begin{gathered}
\left. 1 \right)\left\{ \begin{gathered}
x > 1 \hfill \\
x > 0 \hfill \\
x \ne 1 \hfill \\
{\log _x}125 < {\log _x}{x^3} \hfill \\
\end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered}
x > 1 \hfill \\
x > 0 \hfill \\
x \ne 1 \hfill \\
125 < {x^3} \hfill \\
\end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered}
x > 1 \hfill \\
x > 0 \hfill \\
x \ne 1 \hfill \\
5 < x \hfill \\
\end{gathered} \right. \hfill \\
x \in \left( {5; + \infty } \right) \hfill \\
\hfill \\
\left. 2 \right)\left\{ \begin{gathered}
0 < x < 1 \hfill \\
x > 0 \hfill \\
x \ne 1 \hfill \\
{\log _x}125 < {\log _x}{x^3} \hfill \\
\end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered}
0 < x < 1 \hfill \\
x > 0 \hfill \\
x \ne 1 \hfill \\
125 > {x^3} \hfill \\
\end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered}
0 < x < 1 \hfill \\
x > 0 \hfill \\
x \ne 1 \hfill \\
5 > x \hfill \\
\end{gathered} \right. \hfill \\
x \in \left( {0;1} \right) \hfill \\
\end{gathered} \]

Решења су $x \in \left( {0;1} \right) \cup \left( {5; + \infty } \right)$