Problem 69 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

\(\log _{(1 / 2)} x>\log _{(1 / 3)} x\)

Step-by-Step Solution

Answer

The solution of the inequality is \(x > 1\).

Step 1: Transformation into exponential form

The inequality can be transformed into the corresponding exponential inequality by using the fact that if \(\log_{b}a=c\), then \(b^c=a\). Therefore, you get \((1/2)^\log_{(1/2)} x > (1/3)^\log_{(1/3)} x\) which simplifies to \(x > (1/3)^\log_{(1/3)} x\).

Step 2: Simplification and solution of the inequality

It will be necessary to find the intersection between the two functions \(y=x\) and \(y=(1/3)^\log_{(1/3)} x\). By setting these two equal, you get \(x = (1/3)^\log_{(1/3)} x\). Solving by using the properties of logarithms, you arrive at \(x = 1\).

Step 3: Finding the inequality solution interval

Since the question asks for \(x > (1/3)^\log_{(1/3)} x\), it is clear that \(x\) should be greater than 1. Therefore, the solution of the inequality is \(x > 1\).

Key Concepts

Exponential Inequalities

Exponential inequalities involve comparing expressions with exponential terms. Unlike regular inequalities with constants or linear expressions, exponential inequalities require special attention. In these inequalities, the variable appears as an exponent, which makes them quite different from the usual algebraic inequalities.

• When dealing with exponential inequalities, it is often necessary to manipulate the expression to make the variable apparent.
• An inequality like \(b^x > b^y\) holds when \(x > y\), assuming \(b\) is a positive number different from 1.
• If the base \(b\) is less than 1, the inequality direction reverses: \(b^x > b^y\) implies \(x < y\).

In our exercise, initially converting the inequality into exponential form allows us to apply such principles. This step simplifies the process of finding the solution because we can analyze it terms we understand better.

Properties of Logarithms

Logarithms have unique properties that can transform complex equations into manageable forms. The essential properties allow us to solve logarithmic problems effectively and are instrumental in simplifying logarithmic inequalities.

• Change of Base Rule: The log base change formula, \(rac{\log_b{a}}{\log_b{c}} = \log_c{a}\), enables us to switch bases, making calculations easier.
• Inverses: Logarithms and exponentials are inverse functions, which means \(b^{\log_b{x}} = x\).
• Logarithm of One: For any positive base \(b \, (b eq 1)\), \( ext{log}_b{1} = 0\), because \(b^0 = 1\).

These properties help make complex inequalities solvable by converting logarithmic expressions into exponential ones and vice versa. Knowing these properties supports interpreting problems that initially seem challenging, like the exercise at hand.

Logarithmic Functions

Logarithmic functions, a key component of log inequalities, convert multiplicative relationships into additive ones. These functions are defined with specific bases and convey how one quantity relates to another multiplicatively.

• A logarithmic function with base \(b\) is defined as \(y = \log_b{x}\), meaning \(b^y = x\).
• The base \(b\) must be greater than zero and not equal to one. This ensures the function is well-defined and useful in real-world applications.
• Logarithmic functions graphically appear as curves that increase or decrease but never touch the x-axis, depending on the base.

When dealing with inequalities, understanding the behavior of logarithmic functions helps determine the solutions. For instance, assessing whether the function will grow or shrink aids in interpreting inequality results, useful here in deciding the solution components \(x > 1\). Managing log functions and using them with confidence is vital when handling math problems in various domains.