Problem 307 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \log _{7} x-\log _{x} \frac{1}{7} \geq 2 $$

Step-by-Step Solution

Answer

The short answer to the inequality \(\log_{7} x - \log_{x} \frac{1}{7} \geq 2\) is \(x \geq \sqrt[6]{49}\).

Step 1: Combine the logarithms

We can combine the logarithms by using the property of logarithms mentioned above. In this case, we have: \[ \log_{7} x - \log_{x} \frac{1}{7} = \log_7 x \cdot \log_x \frac{1}{7^{-1}}, \] which simplifies to: \[ \log_7 \left(\frac{x^7}{x}\right) \geq 2. \]

Step 2: Rewrite the inequality in exponential form

Recall that the definition of the logarithm \(\log_b a = c\) is equivalent to the exponential form \(b^c = a\). Applying this, we get: \[ 7^2 \le x^7\cdot \frac{1}{x}. \]

Step 3: Simplify the inequality

Now simplify the inequality as follows: \[ 49 \le x^6. \]

Step 4: Solve for x

Take the sixth root of both sides: \[ \sqrt[6]{49} \le x, \] and notice that \(x\) cannot be negative, since the original equation involves logarithms which are undefined for negative values. So the inequality becomes: \[ x \geq \sqrt[6]{49}. \] Thus, the solution to the inequality is \(x \geq \sqrt[6]{49}\).

Key Concepts

Properties of logarithms

Understanding the properties of logarithms is key to solving logarithmic inequalities. Logarithms are used to determine the exponent needed for a base number to achieve a certain value. Here are some important properties that are especially helpful:

• The product property allows you to combine logarithms that are being added: \( \log_b(MN) = \log_b M + \log_b N \).
• The quotient property helps you work with logarithms that are being subtracted: \( \log_b(\frac{M}{N}) = \log_b M - \log_b N \).
• The power property allows you to simplify logarithms where the value is raised to a power: \( \log_b(M^k) = k \cdot \log_b M \).

In the exercise provided, these properties help convert and simplify the inequality. By recognizing the structure, we can rearrange the expression and make it easier to solve.

Inequality solving

Solving inequalities often requires manipulation similar to solving equations, but with extra attention to the direction of inequality signs. Here are some tips to remember:

• Perform the same mathematical operation on both sides to simplify or solve the inequality.
• When multiplying or dividing both sides by a negative number, remember to flip the inequality sign.
• Allow for variables to take on a range of possible values, ensuring each step respects these possibilities.

In the given exercise, we see these principles in action. The problem was transformed using properties of logarithms, leading to a simpler inequality: \( 49 \le x^6 \). Taking the sixth root on both sides while respecting the non-negative nature of \( x \), results in \( x \geq \sqrt[6]{49} \). This step points out the critical role of understanding how arithmetic operations affect inequalities.

Logarithmic functions

Logarithmic functions are inverse operations of exponential functions and play a significant role in solving equations with variable exponents. Logarithmic functions have a few distinctive features:

• The graph of a logarithmic function \( \log_b x \) passes through the point (1,0), and the function increases if the base \( b \) is greater than 1.
• It is important to note that logarithms are only defined for positive values of \( x \). This restriction is crucial when finding solutions.
• When bases are involved in an inequality, as in the original problem, understanding how logarithmic scales operate can help deduce or constrain solutions.

The exercise effectively shows how crucial it is to think about the domain of the logarithmic function. Because log terms only work for positive \( x \), the solution to the inequality ensures that \( x \) is greater than or equal to the sixth root of 49, thus respecting the logarithm's domain and range constraints.