Problem 17 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The number of real roots of \(x \ln x-1=0\) is (a) 2 (b) 1 (c) 3 (d) infinite

Step-by-Step Solution

Answer

The number of real roots of the given equation is 1. So, the correct answer is (b) 1.

Step 1: Understand the given equation

The given equation is \(x \ln x - 1 = 0\). It involves a product of 'x' and natural logarithm of 'x' and both factors need to be greater than zero.

Step 2: Determine the range for which \(\ln x\) is defined

The natural logarithm \(\ln x\) is defined for \(x > 0\). So, only positive 'x' exists.

Step 3: Check for real roots

For \(x \ln x - 1 = 0\), if we plot the function \(y = x \ln x\), the curve lies above the line \(y = 1\) except for a single point of intersection. This means that the given equation has only one real root.

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, denoted as \(\text{ln}(x)\), is essentially the inverse operation of raising the mathematical constant, Euler's number \(e\), to a given power. In simpler terms, if \(\text{ln}(a) = b\), then you know that \(e^b = a\). The natural logarithm is a vital concept in calculus, and it can be applied across various disciplines such as physics, engineering, and finance.

The function \(\text{ln}(x)\) is only defined for positive real numbers—that is, you cannot find the natural logarithm of a negative number or zero. This peculiarity stems from the properties of the exponential function which does not cover negative numbers or zero when considering real-valued outputs.

When dealing with logarithmic equations, it's important to remember that the argument of the logarithm, in this case 'x', must be greater than zero to obtain a real number. This restriction is essential when trying to determine the domain and the potential solutions for logarithmic equations.

Determining Real Roots

Real roots are the solutions to an equation that are real numbers, as opposed to imaginary or complex numbers. In the context of logarithmic equations, determining the real roots often involves setting the equation to zero and solving for the variable involved.

However, because logarithmic functions only take positive arguments, an added layer of analysis must be performed to ensure that any solutions fall within the acceptable domain. In the exercise at hand, we have \(x \text{ln}(x) - 1 = 0\). Solving for 'x' means finding the value where the function \(x \text{ln}(x)\) crosses the line \(y=1\).

You can apply various strategies such as graphing, numerical approximation, or analytical methods to locate where this crossing happens. The most important thing is to remember that any potential root must result in a positive value of 'x' because \(\text{ln}(x)\) is undefined for non-positive values.

Graphical Analysis of Functions

Graphical analysis is a powerful tool for understanding the behavior of functions, particularly when it comes to finding roots. When you graph a function, you can visually identify points of intersection with the x-axis, which represent the real roots of the equation.

To solve the given exercise graphically, one would plot the function \(y = x \text{ln}(x)\) and then draw the line \(y = 1\). The real roots of the equation correspond to the x-values where the two graphs intersect. Since a logarithmic curve approaches the x-axis infinitely and never touches or crosses it, the number of intersections points with the line \(y = 1\) dictates the number of real roots.

In this exercise, the graphical analysis allows us to verify that the function indeed intersects the line \(y = 1\) at exactly one point. This understanding leads us to conclude that there is only one real root, illustrating how the visual inspection of functions can be a helpful step in the problem-solving process.