Problem 1 - Algebra for JEE ADVANCED

Question

If \(\log _{2}\left(5 \times 2^{x}+1\right), \log _{4}\left(2^{1-x}+1\right)\) and 1 are in A.P., then \(x\) equals a. \(\log _{2} 5\) b. \(1-\log _{5} 2\) c. \(\log _{5} 2\) d. none of these

Step-by-Step Solution

Answer

x = \( \log_{5} 2 \).

Step 1: Understand the Problem

We are given that \( \log_{2}(5 \times 2^{x} + 1), \log_{4}(2^{1-x} + 1), \) and 1 are in Arithmetic Progression (A.P.). In an A.P., the difference between consecutive terms is constant. We need to find the value of \( x \).

Step 2: Express "In A.P." Condition

If three terms \( a, b, c \) are in A.P., then \( 2b = a + c \). Applying this to our logs, we get: \( 2 \log_{4}(2^{1-x} + 1) = \log_{2}(5 \times 2^{x} + 1) + 1 \).

Step 3: Simplify the Equation

Rewrite \( \log_{4}(2^{1-x} + 1) \) as \( \frac{1}{2} \log_{2}(2^{1-x} + 1) \) because \( \log_{4} a = \frac{1}{2} \log_{2} a \). Substituting into our equation gives: \( \log_{2}(2^{1-x} + 1) = \frac{1}{2} \left( \log_{2}(5 \times 2^{x} + 1) + 1 \right) \).

Step 4: Simplify and solve for x

Equating the logarithms from the simplified equation, we have: \( 2^{1-x} + 1 = (5 \times 2^{x} + 1)^{1/2} \cdot 2^{1/2} \). Squaring both sides and simplifying gives: \( (2^{1-x} + 1)^2 = 2 (5 \times 2^{x} + 1) \). Solving this equation yields \( x = \log_{5} 2 \).

Step 5: Verify the Solution

Verify that \( x = \log_{5} 2 \) satisfies the original A.P. condition: Calculate each term and check that the difference remains constant. If it holds true, then this is the solution.

Key Concepts

Logarithms

Logarithms are incredibly useful tools in mathematics for simplifying complex equations and operations, especially when dealing with exponentiation. Essentially, a logarithm answers the question: 'To what power must a base be raised to obtain a given number?' For example, if we say \( \log_{2} 8 = 3 \), it means \( 2^3 = 8 \). Logarithms have several important properties which make them very helpful in solving mathematical problems:

• Product Rule: \( \log_{b}(MN) = \log_{b}M + \log_{b}N \)
• Quotient Rule: \( \log_{b}\left( \frac{M}{N} \right) = \log_{b}M - \log_{b}N \)
• Power Rule: \( \log_{b}(M^n) = n \cdot \log_{b}M \)

These properties allow us to manipulate and transform expressions involving logarithms into simpler forms. In equations, changing the base of a logarithm is often necessary, as shown by the relation \( \log_{b}a = \frac{\log_{c}a}{\log_{c}b} \). This change of base is crucial, especially when the logarithms involved have different bases, like \( \log_{2} \) and \( \log_{4} \). Understanding how to work with these rules helps streamline the process of solving logarithmic equations.

Equation Solving

Solving equations is a fundamental aspect of algebra, requiring the identification of variable values that satisfy the given mathematical expressions. In this exercise, solving the logarithmic and exponential equations involves several steps:

• Simplification: Convert all logarithms to a common base if possible. The problem simplifies \( \log_{4}(...) \) to \( \log_{2}(...) \) using the identity \( \log_{4}a = \frac{1}{2}\log_{2}a \).
• Substitution and Equating: Substitute equivalent expressions to transform the original problem into a simpler form, \( 2b = a + c \) for terms in an A.P., into a single equation.
• Solving for Variables: Once simplified, solve for the unknown variable \( x \). This might involve further algebraic manipulation such as squaring both sides, as practiced in the original problem.

Always double-check your solution by substituting back into the original conditions to verify correctness. This ensures that the solution is not just mathematically correct, but also contextually applicable to the problem.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a mathematical statement. These expressions are central to understanding equations and inequalities in mathematics, including those involving logarithms and exponentials.

• Simplifying Expressions: The process often begins by simplifying expressions, combining like terms, and factoring where possible. It helps reduce the complexity of an equation before solving for unknowns.
• Understanding Exponents: Expressions involving exponents are common; knowing the laws of exponents aids in simplifying and solving tasks. For instance, \( a^m \cdot a^n = a^{m+n} \) and \( (a^m)^n = a^{mn} \).
• Transformations: Using substitutions to transform the equations can make them more manageable. In our problem, rewriting logarithms and applying exponentiation were key steps.

These algebraic manipulations are critical for solving equations neatly and efficiently. In the context of this problem, such skills allow the conversion of complex logarithmic expressions into a linear form that can then be solved arithmetically.