Prove.

Question

Prove.

mathmath333 · Answer

$\large \color{black}{\begin{align} & 1.)\ \normalsize \text{If }\ a>1\ \text{and}\ \log_{a}x_{1}>\log_{a}x_{2}, \hspace{.33em}\~\ & \normalsize \text{then}\ x_{1}>x_{2} \hspace{.33em}\~\ & 2.)\ \normalsize \text{If }\ a<1\ \text{and}\ \log_{a}x_{1}>\log_{a}x_{2}, \hspace{.33em}\~\ & \normalsize \text{then}\ x_{1}

michele_laino · Answer

case 1)
we can write:
$$\Large \begin{gathered}
  {\log _a}{x_1} - {\log _a}{x_2} > 0 \hfill \
  {\log _a}\left( {\frac{{{x_1}}}{{{x_2}}}} \right) > 0 \Rightarrow \frac{{{x_1}}}{{{x_2}}} > 1 \hfill \ 
\end{gathered} $$

michele_laino · Answer

case 2)
we have
$$\Large  \begin{gathered}
  {\log _a}{x_1} - {\log _a}{x_2} > 0 \hfill \
  {\log _a}\left( {\frac{{{x_1}}}{{{x_2}}}} \right) > 0 \Rightarrow \frac{{{x_1}}}{{{x_2}}} < 1 \hfill \ 
\end{gathered} $$

michele_laino · Answer

please keep in mind these graphs:
|dw:1437675202816:dw|