Problem 329 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ x^{\log x}>10 \cdot x^{-\log x}+3 $$

Step-by-Step Solution

Answer

The solution to the inequality \(x^{\log x} > 10 \cdot x^{-\log x} + 3\) is \(0 < x \leq 1\) or \(x > 1\).

Step 1: Simplify the Inequality

First, we can simplify the inequality by factoring out the common \(x^{\log x}\) term from both sides. This renders the equation into a more manageable form: \(x^{\log x}(1 - 10) > 3\)

Step 2: Solve for the variable

Next, divide both sides of the inequality by the term \((1-10)\) to isolate the \(x^{\log x}\) term: \(x^{\log x} > -\frac{3}{9}\)

Step 3: Apply the property of Logarithms

Recall that the property of logarithms holds: \(a^{log_a(b)} = b\). Using this property, the inequality simplifies further to: \(x > x^{-\frac{3}{9}}\)

Step 4: Finding the domain of the inequality

To solve this inequality, note that any real number raised to a negative power is always positive. Hence, the domain of this inequality is \((0, \infty)\)

Step 5: Final solution

Because we are dealing with an inequality and not an equality, we need to identify the subset of the domain for which the inequality holds true. Considering the nature of the function, our final solution set is \((0, 1] \, U \, (1, \, \infty)\)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions are of the form \(y = a^x\) where \(a\) is a positive constant called the base, and \(x\) is the exponent. These functions grow rapidly since the variable \(x\) appears in the exponent.Exponential functions have several important properties:

• The base, \(a\), is always a positive real number.
• If the base \(a = 1\), the function is constant and equal to 1 regardless of \(x\).
• As the exponent \(x\) increases, the function value \(y = a^x\) also increases when \(a > 1\).
• For bases \(0 < a < 1\), the function decreases as the exponent \(x\) increases.

Exponential functions are widely used in real-world situations like population growth, radioactive decay, and interest calculations. In solving inequalities involving exponential functions, it's important to identify a common base and apply logarithmic properties to simplify the problem.

Logarithmic Properties

Logarithms are the inverse operations of exponentials, helping to express powers. When dealing with complex expressions involving exponents, logarithms can simplify calculations considerably. For any positive number \(a\) (where \(a eq 1\)), the logarithm of a number \(x\) is the exponent to which the base \(a\) must be raised to obtain \(x\). This is denoted as \(y = \log_a{x}\) if and only if \(a^y = x\). Some key properties of logarithms include:

• \(\log_a(1) = 0\) because \(a^0 = 1\).
• \(\log_a(a) = 1\) because \(a^1 = a\).
• \(\log_a(xy) = \log_a(x) + \log_a(y)\)
• \(\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\)
• \(\log_a(x^n) = n \cdot \log_a(x)\)

Understanding these properties allows us to transform and solve equations or inequalities in a way that simplifies the calculations. Applying these logarithmic rules in the given problem aids in expressing powers in equivalent forms.

Domain of Functions

The domain of a function is the complete set of possible values of the independent variable that makes the function work without any errors such as division by zero or negative square roots. For exponential functions, the domain is typically all real numbers because the base \(a\) is greater than zero.In some cases, the domain of the solution to a problem can differ from the domain of the involved functions. For example, while the domain of \(x^{\log x}\) is all positive real numbers \((0, \infty)\), the solution to an inequality containing this function may be a subset of that domain.To determine the domain of a specific inequality, such as in this exercise, it is crucial to identify restrictions and apply logical reasoning. For instance, if the equation deals with negative powers, as with \(x^{-\log x}\), it's useful to remember:

• Powers must remain defined and real.
• We cannot calculate logarithms for non-positive numbers.

By carefully evaluating these criteria, we achieve a valid domain for solution: in this case, \((0, 1] \cup (1, \infty)\), ensuring all operations are mathematically valid and meaningful.