Problem 21 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\log _{3} 2, \log _{3}\left(2^{x}-5\right), \log _{3}\left(2^{x}-\frac{7}{2}\right)\) are in A.P., then find the value of \(x\)

Step-by-Step Solution

Answer

The value of \(x\) that makes the expressions an arithmetic progression is \(x = 3\)

Step 1: Formulate the Equation

The condition that the three terms are an arithmetic progression can be translated into a mathematical equation. In an A.P., the difference between any two successive terms is a constant. So, \(\log _{3}\left(2^{x}-5\right) - \log _{3} 2 = \log _{3}\left(2^{x}-\frac{7}{2}\right) - \log _{3}\left(2^{x}-5\right)\)

Step 2: Simplify the Equation using Logarithm Properties

From the properties of logarithms, the difference between the log of two numbers is the log of the quotient. Apply this property to simplify each side of the above equation. So, the equation becomes \(\log _{3}\left(\frac{2^{x}-5}{2}\right) = \log _{3}\left(\frac{2^{x}-\frac{7}{2}}{2^{x}-5}\right)\).

Step 3: Apply Equal Logarithms Property

The only way the two sides of the equation can be equal is if the arguments of the logarithms are equal. So, we can say that \(\frac{2^{x}-5}{2} = \frac{2^{x}-\frac{7}{2}}{2^{x}-5}\). Simplify this equation to find \(x\).

Step 4: Solve for the variable \(x\)

\(\frac{2^{x}-5}{2} = \frac{2^{x}-\frac{7}{2}}{2^{x}-5}\) can be simplified and solved for \(x\) resulting in \(x = 3\)

Key Concepts

Logarithmic Equations

When dealing with logarithmic equations, we often need to find unknowns like the variable in a problem. A logarithmic equation typically involves the logarithms of expressions that incorporate the variable.
The example in the exercise uses three logarithmic expressions: \(\log_3 2\), \(\log_3\left(2^x-5\right)\), and \(\log_3\left(2^x-\frac{7}{2}\right)\). These are sequenced in an arithmetic progression (A.P.). This means their differences are consistent. Therefore, the main task is to leverage this property to form a solvable equation.
Key steps involve evaluating and setting equal terms that mirror typical arithmetic operations performed with logarithms. Understanding the base of logarithms (in this case, base 3) and applying properties enables us to simplify and solve the problem.

Properties of Logarithms

Properties of logarithms are the backbone of simplifying complex logarithmic expressions. Here are the crucial properties:

• Logarithm of a Quotient: \(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\)
• Logarithm of a Product: \(\log_b (MN) = \log_b M + \log_b N\)
• Logarithm of a Power: \(\log_b (M^n) = n \cdot \log_b M\)

In this exercise, the property of the logarithm of a quotient is particularly useful. We shift from subtracting two logarithmic terms to expressing the ratio of their arguments.
This property allows transforming the initial expressions from differences into equatable logarithmic expressions, hence simplifying the problem dramatically.

Solving Equations

Once we have the simplified logarithmic expressions, solving the equation plays out as straightforward algebra. After applying logarithmic properties, we simplify the scenario to a basic algebraic equation where we equate the expressions being logged.
In the equation part, \(\frac{2^x-5}{2} = \frac{2^x-\frac{7}{2}}{2^x-5}\), it's crucial to cross-multiply to eliminate fractions. This simplification leads us to a manageable expression that can be treated with basic algebraic operations.
Simplification step-by-step:

• Cross-multiply to eliminate denominators.
• Solve the resulting equation to isolate and derive \(x\).

This method yields \(x = 3\), showcasing how transforming logarithmic equations into simplified algebraic terms leads us to the solution.