Logarithmic Inequalities: Definition, Problems with Solutions

Inequalities are mathematical expressions showing the relationship between two values, indicating that one value is greater than, less than, or not equal to another. Understanding inequalities is crucial for solving various mathematical problems, from basic arithmetic to advanced calculus. Especially, logarithmic inequalities are important in solving log-related inequalities.

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In this article, we will cover the concepts of logarithmic inequalities. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of two questions have been asked on this concept, including two in 2023.

Inequalities

Inequalities are the relationship between two expressions that are not equal to one another. Symbols denoting the inequalities are $<,>,\le ,\ge$, and $\ne$.
- $x<4$, "is read as $x$ less than 4 ", $x\le 4$, "is read as $x$ less than or equal to 4 ".
- Similarly $x>4$, "is read as $x$ greater than 4 " and $x\ge 4$, "is read as $x$ greater than or equal to $4^{\text{"N}}$.

The process of solving inequalities is the same as of equality but instead of equality symbol inequality symbol is used throughout the process.

Logarithmic inequalities:

$\begin{array}{rl}& {\log }_{a}x>{\log }_{a}y=\{\begin{array}{lc}x>y,& \text{if}a>1\\ x<y,& \text{if}0<a<1\end{array}\\ & {\log }_{a}x>y=\{\begin{array}{ll}x>a^y,& \text{if}a>1\\ x<a^y,& \text{if}0<a<1\end{array}\\ & {\log }_{a}x<y=\{\begin{array}{ll}x<a^y,& \text{if}a>1\\ x>a^y,& \text{if}0<a<1\end{array}\end{array}$So, we change the sign of inequality while removing log when the base is less than 1

Note: We always take the intersection of the answer of this inequality with the domains of all the log terms involved in the inequality.

Summary

We concluded that inequalities are a fundamental part of mathematics, providing a way to describe and solve problems involving ranges and constraints. Mastery of inequalities is essential for progressing in algebra, calculus, and applied mathematics, offering valuable tools for both theoretical and practical problem-solving.

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Solved Examples Based On the Logarithmic Inequalities:

Example 1: What is the value of $x$ satisfying the inequality ${\log }_5(x+5)>{\log }_7(x+5)$ ?
Solution:
First, we change the base to make the base of both sides the same.

$\begin{array}{rl}& {\log }_{b}x=\frac{{\log }_{a}x}{{\log }_{a}b}\\ & \frac{\log (\mathrm{x}+5)}{\log 5}>\frac{\log (\mathrm{x}+5)}{\log 7}\end{array}$

Now for this equation to be true, we observe that

$\begin{array}{rl}& \frac{\log (x+5)}{\log 5}>\frac{\log (x+5)}{\log 7}\\ & \frac{\log (x+5)}{\log (5)}-\frac{\log (x+7)}{\log 7}>0\\ & \log (x+5)>0\\ & x+5>{10}^0\\ & x+5>1\\ & x>-4\end{array}$

Example 2: The number of distinct solutions of the equation, ${\log }_{\frac{1}{2}}|\sin x|=2-{\log }_{\frac{1}{2}}|\cos x|$ in the interval $[0,2\pi ]$, is $$
Solution:

$\begin{array}{rl}& {\log }_{\frac{1}{2}}(|\sin (x)|)+{\log }_{\frac{1}{2}}(|\cos (x)|)=2\\ & {\log }_{\frac{1}{2}}(|\sin (x)||\cos (x)|)=2\\ & {\log }_{\frac{1}{2}}(|\sin (x)||\cos (x)|)={\log }_{\frac{1}{2}}\left(\frac{1}{4}\right)\\ & |\sin x||\cos x|=\frac{1}{4}\\ & \sin 2\theta =\frac{1}{2}\\ & x=\frac{\pi }{12}+\pi n,x=\frac{5\pi }{12}+\pi n\end{array}$
The total number of solutions is 8

Hence, the answer is 8.

Example 3 Let a complex number $z,|z|\ne 1$, satisfy ${\log }_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1{)}^2}\right)\le 2$ Then, the largest value of $|z{|}_{\text{is equal to}}$ $$
Solution:

$\begin{array}{r}{\log }_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1{)}^2}\right)\le 2\\ \frac{|z|+11}{(|z|-1{)}^2}\ge \frac{1}{2}\\ 2|z|+22\ge (|z|-1{)}^2\\ 2|z|+22\ge |z{|}^2+1-2|z|\\ |z{|}^2-4|z|-21\le 0\\ (|z|-7)(|z|+3)\le 0\\ |z|\in [-3,7]\end{array}$

$\therefore$ Largest value of $|z|$ is 7
Hence, the answer is 7 .

Hence, the answer is 7.

Example 4: The solution set of the inequation $1+{\log }_{\frac{1}{3}}(x^2+x+1)>0$ is:
Solution:

$\begin{array}{rl}& \text{(i)}{\log }_{\frac{1}{3}}(x+x^2+1)>-1\\ & \Rightarrow x^2+x+1<3\\ & \Rightarrow x^2+x-2<0\\ & \Rightarrow (x+2)(x-1)<0\Rightarrow x\in (-2,1)\\ & \text{and}\end{array}$

(ii) $x^2+x+1>0\Rightarrow x\in R$. $$ (ii)
by (i) $\mathrm{\&}$ (ii) $x\in (-2,1)$

Example 5: What is the value x when ${\log }_{\frac{1}{2}}(x-1)>1$
Solution:

$\begin{array}{rl}& {\log }_{0.5}(x-1)>1\\ & (x-1)<(0.5{)}^1\\ & x-1<0.5\text{and}x-1>0\\ & 1<x<1.5\end{array}$

Frequently Asked Questions(FAQ)-
1. What are inequalities?

Ans: Inequalities are the relationship between two expressions that are not equal to one another.
2. Greatest integer less than or equal to the number ${\log }_{2}15.{\log }_{1/6}2{\log }_{3}1/6$ is:

Ans:$\begin{array}{rl}& {\log }_{2}15\cdot {\log }_{1/6}2.{\log }_{3}1/6=\frac{\log 15}{\log 2}\times \frac{\log 2}{\log 1/6}\times \frac{\log 1/6}{\log 3}\\ & =1+{\log }_{3}5>2\end{array}$

3. What is a logarithmic inequality?

Ans: A logarithmic inequality is an inequality that involves a logarithmic function, such ${}^s{\log }_b(x)$, where b is the base of the logarithm.
4. How do you solve logarithmic inequalities with different bases?

Ans: When dealing with logarithmic inequalities with different bases, it is often useful to convert them to the same base using the change of base formula