Logarithmic Inequalities

Solving logarithmic inequalities, it is important to understand the direction of the inequality changes if the base of the logarithms is less than 1.
$$\log_{2}{x} \lt \log_{2}{y}, \text{ then } x \lt y \\
\log_{0.5}{x} \lt \log_{0.5}{y}, \text{ then } x \gt y \\
$$
Also the domain of the logarithm is positive.
$$\log_{10}{(x-2)}, \text{ then } x-2 \gt 0$$

Question 1

Solve \( \log_{3}{(x-3)} \gt \log_{3}{(x-1)}. \)

Question 2

Solve \( \log_{3}{(x-3)} \gt \log_{9}{(x-1)}. \)

Question 3

Solve \( \log_{0.5}{(x^2-19)} – \log_{0.5}{(x-5)} \lt \log_{0.5}{5} \).

Question 4

Solve \( (\log_{3}{x})^2 \lt \log_{3}{x^4} \).

Question 5

Solve \( x^{\log_{2}{x}} \lt 8 x^2 \).