Problem 66 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

\(\log _{(1 / 2)}\left(\log _{5}\left(\log _{2}\left(x^{2}-6 x+40\right)\right)\right)>0\)

Step-by-Step Solution

Answer

The solution to the inequality is \(2 < x < 4\). All x-values within this interval would satisfy the inequality.

Step 1: Dealing with the outermost logarithm inequality

Firstly, notice that the base of the outermost logarithm is \(1/2\), which is less than 1, so the inequality sign changes direction when we remove the outermost logarithm. This inequality can be written as: \(\log_{5}(\log_{2}(x^{2}-6x+40)) < 1\)

Step 2: Removing the second logarithm

The base of the second logarithm is 5, so remove this logarithm without changing the inequality direction: \(\log_{2}(x^{2}-6x+40) < 5^1 = 5\)

Step 3: Removing the third and final logarithm

For the third logarithm, the base is 2. By applying the properties of logarithms, the inequality can be transformed into: \(x^{2}-6x+40 < 2^5 = 32\)

Step 4: Solving the quadratic inequality

To solve this quadratic equation, subtract 32 from both sides: \(x^{2}-6x+40 -32 < 0\), or \(x^{2}-6x+8 < 0\). Factoring this gives: \((x-2)(x-4)<0\)

Step 5: Finding the solutions

According to the properties of inequalities, the solution to this inequality would lie between the two roots, 2 and 4. Testing a number that lies between these roots, such as 3, in the original inequality will confirm this.

Key Concepts

Logarithmic Inequalities

Logarithmic inequalities involve comparisons that include logarithmic expressions. Understanding how to solve them hinges on several key properties of logarithms.

• Firstly, remember that if you have a logarithm like \log_b(a), its outcome depends on the base \(b\). When \(b > 1\), increasing \(a\) increases the log value. But when \(0 < b < 1\), increasing \(a\) decreases the log result.
• One crucial rule is that with bases less than 1, inequality signs flip direction when you remove the logarithm. For example, from \(\log_{1/2}(a) > 0\), you infer \(a < 1\).
• To solve logarithmic inequalities, you often start by removing one logarithm at a time, using its properties to preserve or reverse the inequality sign.

Returning to our exercise, we began solving by simplifying the outer logarithmic inequality because its base, \(\frac{1}{2}\), is less than 1, requiring us to reverse the inequality direction upon removing the logarithm.

Quadratic Inequalities

Once the logarithmic steps simplify to a polynomial inequality, you'll encounter quadratic inequalities. A quadratic inequality involves terms like \(ax^2 + bx + c < d\). Solving these requires different strategies compared to linear inequalities.

• The first step is to express the inequality in standard form by moving all terms to one side: \(ax^2 + bx + (c - d) < 0\).
• Next, find the roots of the corresponding quadratic equation \(ax^2 + bx + c = 0\) using methods like factoring, the quadratic formula, or completing the square.
• Once you have the roots, analyze the intervals they create to find where the inequality holds. You're checking where the graph of the quadratic dips below the x-axis.

Our exercise illustrated this: First, we factored \(x^2 - 6x + 8\) to \((x-2)(x-4) < 0\), and determined that the inequality held true between roots 2 and 4.

Solving Inequalities

Solving inequalities means finding all possible values of the variable that satisfy the given condition. Unlike equations, inequalities offer ranges of solutions rather than fixed values.

• Begin by performing operations to isolate the variable on one side. This might involve adjusting terms across the inequality to simplify.
• Any time you multiply or divide by a negative number, flip the inequality sign. This is a critical step to remember.
• Graphical understanding can be beneficial as it lets you visualize where a polynomial is greater or less than a certain value.

In our exercise, after breaking down the logarithmic and quadratic steps, we graphically determined the solution by testing values and ensuring our interval (2, 4) delivered consistent results. Thus, recognizing the solution involves both algebraic manipulation and visualization.