Tag Archives: Inequality

Exponential Inequalities using Logarithms

Exponential Inequalities using Logarithms

Inequalities worked in the same way, except there was a change of sign when dividing or multiplying both sides of the inequality by a negative number. \begin{array}{|c|c|c|} \hline\log_{2}{3}=1.6>0 & \log_{5}{3}=0.7>0 & \log_{10}{3}=0.5>0 \\ \hline\log_{2}{2}=1>0 & \log_{5}{2}=0.4>0 & \log_{10}{2}=0.3>0 \\ \hline\log_{2}{1}=0 & \log_{5}{1}=0 & \log_{10}{1}=0 \\ \hline\log_{2}{0.5}=-1<0 & \log_{5}{0.5}=-0.4<0 & \log_{10}{0.5}=-0.3<0 \\ \hline\log_{2}{0.1}=-3.3<0 & \log_{5}{0.1}=-1.4<0 & […]

Logarithmic Inequalities

Logarithmic Inequalities

In 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 […]

Inequality with Variables in Denominator

Inequality with Variables in Denominator

Solving Inequality with Variables in the Denominator requires special care due to the direction of the inequalities. Let’s have a look at the following key points. Key Point 1 \( \begin{aligned} \displaystyle \require{color}\frac{1}{x} &\ge 2 \\\frac{1}{x} \times x &\ge 2 \times x &\color{green} \text{Many of you may think this is TRUE.} \\&&\color{green} \text{This is TRUE […]