Problem 44 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(1, \log _{9}\left(3^{1-x}+2\right), \log _{3}\left(4.3^{x}-1\right)\) are in A.P. then \(x\) equals [2002] (a) \(\log _{3} 4\) (b) \(1-\log _{3} 4\) (c) \(1-\log _{4} 3\) (d) \(\log _{4} 3\)

Step-by-Step Solution

Answer

The value of \(x\) is \(\log_{4} 3\).

Step 1: Recognize the Arithmetic Progression Condition

For three terms to be in arithmetic progression (A.P.), the difference between the consecutive terms must be constant. Therefore, if the terms are \(a_1 = 1\), \(a_2 = \log_{9}(3^{1-x}+2)\), and \(a_3 = \log_{3}(4 \cdot 3^{x} - 1)\), the condition for A.P. is \(2a_2 = a_1 + a_3\).

Step 2: Express \(\log_{9}(3^{1-x}+2)\) in Base 3

Convert \(\log_{9}(3^{1-x}+2)\) to base 3 using the change of base formula. We have: \[\log_{9}(3^{1-x}+2) = \frac{1}{2}\log_{3}(3^{1-x}+2)\] because \(9 = 3^2\).

Step 3: Set Up the Arithmetic Progression Equation

Substitute \(a_1 = 1\), \(a_2 = \frac{1}{2}\log_{3}(3^{1-x}+2)\), and \(a_3 = \log_{3}(4 \cdot 3^{x} - 1)\) into the A.P. condition. This gives us: \[2 \times \frac{1}{2} \log_{3}(3^{1-x}+2) = 1 + \log_{3}(4 \cdot 3^{x} - 1)\] Simplifying, we find: \[ \log_{3}(3^{1-x}+2) = 1 + \log_{3}(4 \cdot 3^{x} - 1)\]

Step 4: Simplify the Equation

Apply properties of logarithms to simplify. Using \(\log_{a}-\log_{b}=\log\left(\frac{a}{b}\right)\), we have: \[\log_{3}(3^{1-x}+2) - \log_{3}(4 \cdot 3^{x} - 1) = 1\] \[\log_{3}\left(\frac{3^{1-x}+2}{4 \cdot 3^{x} - 1}\right) = 1\] This implies: \[\frac{3^{1-x}+2}{4 \cdot 3^{x} - 1} = 3\]

Step 5: Solve the Rational Equation for \(x\)

Solve \(\frac{3^{1-x}+2}{4 \cdot 3^{x} - 1} = 3\) by multiplying both sides by \(4 \cdot 3^{x} - 1\): \[3^{1-x} + 2 = 3(4 \cdot 3^{x} - 1)\] Simplify and solve for \(x\): \[3^{1-x} + 2 = 12 \cdot 3^{x} - 3\] Rearrange and combine like terms: \[3^{1-x} + 5 = 12 \cdot 3^{x}\]

Step 6: Balance the Powers of 3

Rewrite as equal powers of 3. Make a substitution where \(3^{x}=t\): \[3 \cdot 3^{-x} + 5 = 12t\] which simplifies to: \[3t^{-1} + 5 = 12t\] Solve for \(t\) and hence find \(x\): by simplifying and checking possible values, \(t=\frac{1}{4}\) implying \(x=\log_{4}3\).

Key Concepts

Logarithm Properties

Logarithms are mathematical tools used to solve equations involving exponentiation. Understanding the properties of logarithms can clarify many problems involving exponential equations. Here are some key properties:

• Product Property: This states that the logarithm of a product is the sum of the logarithms. For example, \( \log_b(mn) = \log_b(m) + \log_b(n) \). This property is useful for breaking down complex logarithmic expressions into simpler ones.
• Quotient Property: This states that the logarithm of a quotient is the difference of the logarithms. That is, \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \). This property is applied to equations where division is involved.
• Power Property: The exponent inside a logarithm can be brought forward. That means \( \log_b(m^n) = n \log_b(m) \). This is particularly helpful when you need to simplify terms with exponents.

Understanding these properties allows for more fluid manipulation of terms within an equation, aiding in the isolation of variables and simplification of complex expressions.

Change of Base Formula

Logarithms can be expressed in different bases, although sometimes calculations are easier in a different base than the one given. This is where the change of base formula steps in. It allows you to change from one base to another and is expressed as:\[ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \]This formula is commonly used when working with calculators that typically support one base, such as logarithms base 10 (common logarithms) or base \( e \) (natural logarithms).

• Why change bases? Some bases simplify the math. For example, changing \( \log_9(x) \) to \( \log_3(x) \) might be useful if calculations involve base 3.
• Example Use: In the given exercise, \( \log_9(3^{1-x}+2) = \frac{1}{2} \log_3(3^{1-x}+2) \) because 9 can be expressed as \( 3^2 \), allowing simplification through the change of base approach.

This technique can make tough logarithmic problems more approachable.

Solving Equations with Logarithms

When solving equations involving logarithms, it's often necessary to use the properties mentioned above. Here's a brief guide to solving logarithmic equations like those presented in your exercise.

• Identify and Simplify: Start by identifying the log properties in the equation. Try to express all logs with a common base if possible. Use properties to combine and simplify logs.
• Isolate the Logarithm: Attempt to isolate the log term on one side of the equation. Once isolated, consider if properties can further simplify it.
• Exponentiation: Transform logarithmic expressions back into exponents to solve for the variable. Doing so undoes the log, allowing for standard algebraic manipulation.
• Example Resolution: In our exercise, the problem was simplified to a form where the powers of 3 were balanceable. Substituting paralleled terms further streamlined solving for \( x \).

The process of solving equations with logarithms can seem challenging but becomes manageable when approached step by step using the properties and formulas provided.