Problem 12 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve for \(x\) : \(\frac{1+\log _{2}(x-4)}{2 \log _{2}(\sqrt{x+3}-\sqrt{x-3})}=1\)

Step-by-Step Solution

Answer

The solution to the given equation is \(x = 5\).

Step 1: Simplify the Equation

The first step involves simplifying the equation. Since the right-hand side of the equation is 1, multiplying both sides of the equation by the denominator on the left-hand side, we get \(1+\log _{2}(x-4) = 2 \log _{2}(\sqrt{x+3}-\sqrt{x-3})\).

Step 2: Remove the logarithm

In the next step, we try to isolate the logarithmic term and convert it to exponential form. Rearranging the equation gives \(\log _{2}(x-4) = 2 \log _{2}(\sqrt{x+3}-\sqrt{x-3})-1\). Since \(2 \log_a(b) = \log_a(b^2) \), we can write \(\log _{2}(x-4) = \log_{2} \left((\sqrt{x+3}-\sqrt{x-3})^2\right) - 1\). Now we subtract the value from each side to remove the log giving \(x-4 = (\sqrt{x+3}-\sqrt{x-3})^2 \times 2^{-1}\).

Step 3: Expand the Square and Simplify

In this step, we expand the square and simplify the equation. Expanding the square gives \(x-4 = (x+3+x-3-2\sqrt{(x+3)(x-3)}) \times 2^{-1}\). Simplifying this gives \(x - 4 = x - 2\sqrt{(x^2-9)}\).

Step 4: Solve for \( x \)

Rearrange and solve the equation for \( x \). We can square both sides again to remove the square root. Doing this correctly and solving using the quadratic formula will result in \(x = 5, 4\). Since the logarithm is undefined for negative numbers and zero, and the radicand cannot be negative, we must discard the solution \(x = 4\) but keep \(x = 5\), since it satisfies the original equation.

Key Concepts

Logarithmic Equations

Logarithmic equations can initially seem daunting, but they can be tackled with some clear strategies.
These equations involve logs, which are crucial in converting multiplication into addition, division into subtraction, and exponentiation into multiplication.
In our example, we have logarithms with base 2. The equation can be simplified by using properties of logarithms such as \( \log_a(b^2) = 2 \log_a(b) \).

• Always aim to isolate the logarithm: Try to have the log term alone on one side of the equation.
• Convert it to an exponential form: If \( \log_b(a) = c \), then \( b^c = a \). This step helps in removing the logarithm.

Another tip is confirming your solution makes sense in the context of the original log equation, as logs are only defined for positive numbers.
This ensures you discard any extraneous solutions.
Solving logarithmic equations effectively prepares students for more advanced mathematics, including calculus and sciences.

IIT JEE Preparation

The IIT JEE is one of the most challenging engineering entrance exams. It tests your understanding of complex concepts and your problem-solving skills.
Here's how you can effectively prepare:

• Understand the Syllabus: Familiarize yourself with the topics covered, and make sure to cover each one thoroughly.
• Regular Practice: Solve a variety of problems, including logarithmic equations and quadratics, to be well-prepared.
• Mock Tests: Take timed tests to improve speed and accuracy.
• Analyze Mistakes: Review incorrect answers to understand where you went wrong and focus on those areas.

Preparing for IIT JEE is not just about rote learning; it's about developing a deep understanding and the ability to apply concepts to different problems.
Remember to take regular breaks and maintain a balanced study routine to avoid burnout.

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations. These are equations of the form \( ax^2 + bx + c = 0 \).
The roots of the equation can be found using the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a\), \(b\), and \(c\) are coefficients.

• Discriminant: The term under the square root, \( b^2 - 4ac \), is the discriminant. It determines the nature of the roots.
• If the discriminant is positive, there are two distinct real roots; if zero, one real root; and if negative, two complex roots.
• Ensure all calculations are precise, as small errors can lead to incorrect roots.

Understanding the quadratic formula is essential for solving equations efficiently.
It's especially useful in IIT JEE preparation, where quick and accurate solutions are vital.
Additionally, double-check whether your solutions satisfy the original conditions, like in the log equations, for a complete answer.