Logarithmic Inequalities

Definition: Logarithmic Inequality

A (real) logarithmic inequality is an inequality which contains variables as part of the base or the argument of a logarithm.

Algorithm: Solving Inequalities of the Form ${\log }_{a}f(x)>{\log }_{a}g(x)$

We are given the following inequality:

$${\log }_{a}f(x)>{\log }_{a}g(x)$$

Solutions:

• If $0<a<1$, then $\{\begin{array}{l}f(x)<g(x)\\ f(x)>0\end{array}$
• If $a>1$, then $\{\begin{array}{l}f(x)>g(x)\\ g(x)>0\end{array}$

Algorithm: Solving Inequalities of the Form ${\log }_{b(x)}f(x)\ge {\log }_{b(x)}g(x)$

We are given the following inequality:

$${\log }_{b(x)}f(x)\ge {\log }_{b(x)}g(x)$$

Solutions:

$$\{\begin{array}{l}0<b(x)<1\\ f(x)\le g(x)\\ f(x)>0\end{array}\bigcup \{\begin{array}{l}b(x)>1\\ f(x)\ge g(x)\\ g(x)>0\end{array}$$