Problem 20 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\left|1-\log _{1 / 5} x\right|+2=\left|3-\log _{1 / 5} x\right|\), then \(x\) is (a) 2 (b) 5 (c) 1 (d) 3

Step-by-Step Solution

Answer

(b) 5

Step 1: Define the Absolute Values

Split the equation into two cases by removing the absolute value: \nCase 1: \(1-\log_{1/5}x + 2 = 3- \log_{1/5}x\)\nCase 2: \(1-\log_{1/5}x + 2 = - (3-\log_{1/5}x)\)

Step 2: Simplifying Case 1

For Case 1, we simplify the equation, rearranging the terms to isolate \(x\)\nThis gives us \(\log_{1/5}x = -1\)\nSolving the above equation for \(x\) we get \(x = (1/5)^{-1} = 5\)

Step 3: Simplifying Case 2

For Case 2, simplifying again we get \(\log_{1/5}x = 2\)\nSolving this equation for \(x\) we get \(x = (1/5)^{2} = 1/25\)

Step 4: Validation of the Solutions

Substitute the found solutions back into the original equation to verify. Only \(x=5\) satisfies the given equation thus it is the only valid solution

Key Concepts

Understanding Absolute Value Equations

An absolute value equation is an equation in which the unknown variable appears inside an absolute value sign. The absolute value of a number represents its distance from zero on the number line, regardless of direction, making it always non-negative. Absolute value equations often have two solutions because both positive and negative values could satisfy the original equation when the absolute value signs are removed.

For example, given the equation \( |x| = a \), where \( a \) is a positive number, the solutions are \( x = a \) and \( x = -a \). This concept is essential in solving problems like the one in the IIT JEE exam that involve absolute value expressions. To solve such equations, we consider all possible cases that can result from removing the absolute value sign, leading to a set of individual equations to solve.

Logarithm Properties

Understanding the properties of logarithms is crucial when working with logarithmic equations. Some critical properties include:

• The product rule, which states that \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• The quotient rule, \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \).
• The power rule, which informs us that \( \log_b(m^k) = k\log_b(m) \).
• Change of base formula that allows you to rewrite a logarithm in terms of logarithms with another base.
• Lastly, the understanding that if \( \log_b(x) = y \), then by definition, \( b^y = x \).

Grasping these properties allows students to simplify and manipulate logarithmic expressions and is essential for solving the logarithmic equations often encountered in competitive exams such as the IIT JEE.

Solving Logarithmic Equations

To solve logarithmic equations, you need to apply the logarithm properties to isolate the logarithmic part of the equation and then convert it to its exponential form, which can be more straightforward to solve. This process often involves the following steps:

1. Simplify the equation using logarithmic properties.
2. Isolate the logarithmic term.
3. Rewrite the logarithmic equation in its equivalent exponential form.
4. Solve the resulting equation.
5. Check all potential solutions in the original logarithmic equation to validate them, as some might not be valid when substituted back into the logarithm.

For the IIT JEE logarithmic problem provided, the solution involved applying the properties of logarithms to simplify the equations obtained after considering the absolute value components. Verification is paramount to ensure that solutions meet the original equation, especially since logarithms are not defined for negative numbers and zero. Hence, extraneous solutions are possible and need to be discarded.