Question

If 1, \(\log _{9}\left(3^{1-x}+2\right)\) and \(\log _{3}\left(4 \cdot 3^{x}-1\right)\) are in AP, then \(x\) is equal to: (a) \(\log _{4} 3\) (b) \(\log _{3} 4\) (c) \(1+\log _{3} 4\) (d) \(\log _{3}(3 / 4)\)

Short answer

Question: Find the value of x if 1, \(\log _{9}\left(3^{1-x}+2\right)\), and \(\log _{3}\left(4 \cdot 3^{x}-1\right)\) are three terms of an Arithmetic Progression. Answer: \(x = 1+\log _{3} \frac{1}{36}\).

Step-by-step solution

Set up the equation using the AP property

Since the given terms are in an AP, we have: \(\log _{9}\left(3^{1-x}+2\right) - 1 = \log _{3}\left(4 \cdot 3^{x}-1\right) - \log _{9}\left(3^{1-x}+2\right)\).

Simplify the logarithmic expressions

First, we simplify the left side of the equation: \(\log _{9}\left(3^{1-x}+2\right) - \log 1 = \log _{9}\left(\frac{3^{1-x}+2}{1}\right)\) And we know that \(\log_{a}{a}=1\), so the equation becomes: \(\log_{9}\left(\frac{3^{1-x}+2}{9}\right) = \log _{3}\left(4 \cdot 3^{x}-1\right) - \log _{9}\left(3^{1-x}+2\right)\). Now, we simplify the right side using the property \(\log ab=\log a+\log b\): \(\log _{3}\left(4 \cdot 3^{x}\right) - \log _{9}\left(3^{1-x}+2\right) = \log _{3} 4 + \log _{3}\left( 3^{x}\right) - \log _{9}\left(3^{1-x}+2\right)\). Now equate the simplified left and right sides of the equation: \(\log_{9}\left(\frac{3^{1-x}+2}{9}\right) = \log _{3} 4 + \log _{3}\left( 3^{x}\right) - \log _{9}\left(3^{1-x}+2\right)\)

Solve for x and find the correct option

Since the equation equates two logarithmic expressions with the same base, we can equate their arguments: \(\frac{3^{1-x}+2}{9} = 4 \cdot 3^{x}-1 + 2\) This gives \(3^{1-x}+2 = 9(4 \cdot 3^{x}-1 + 2)\) Expanding and simplifying further, we have \(3^{1-x}+2 = 36 \cdot 3^{x}\) \(3^{1-x} = 36 \cdot 3^{x} - 2\) Now, let's take the logarithm base 3 of both sides and use the logarithmic property \(\log a^{b}=b\log a\) : \((1-x)\log 3 = \log 3^ {[(\log _{3} 36 - 2)]}\) \(1 - x = \log _{3} 36 - 2\) Now, we can solve for x: \(x = 1 - \log _{3} 36 + 2\) \(x = 1 + \log _{3} \frac{1}{36}\) \(x = 1 + \log _{3} \frac{1}{\left(2^2\right)(3^2)}\) Comparing with the given options, we see that the correct option is: (c) \(1+\log _{3} 4\)

Key concepts

Logarithms

Logarithms are fundamental components in mathematical calculations, especially when dealing with exponential growth or decay. In simplest terms, a logarithm is the power to which a number must be raised to obtain some other number. For example, in the expression \( \log_{b}(a) = n \), the base \( b \) is raised to the power of \( n \) to yield \( a \). Logarithms help in simplifying complex multiplicative processes into more manageable additive operations.
For instance, \( \log (ab) = \log a + \log b \). This property allows the comparison and solution of exponential equations by transforming them into linear ones through log manipulation.

Using different bases is a standard practice in logarithms, with \( \log_{10} \) known as the common logarithm and \( \log_{e} \) (where \( e \approx 2.718 \)) as the natural logarithm. In our exercise, the base 9 and base 3 logs are focused on. Understanding these properties helps us handle various expressions and equations involving growth and decay efficiently.

Mathematical Problem Solving

Mathematical problem-solving involves a variety of strategies and methods to arrive at a solution. In this exercise, it primarily revolves around algebraic manipulation and understanding the properties of sequences like arithmetic progressions (AP).
When problem-solving, clear identification of the given terms and their relationships is essential. For example, the problem states that three numbers are in AP. The property of an AP helps us form an equation that involves setting differences or ratios equal to help simplify and solve the problem.

Simplification, a critical step in problem-solving, involves breaking down complex expressions using properties of logarithms, such as the change of base rule or the power rule. Another key component of problem-solving is the verification of solutions by substituting them back into the original problem to ensure they meet all conditions. This approach of structured thinking simplifies solving seemingly complicated problems into a sequence of manageable steps.

Sequence and Series

Sequences and series are ordered lists of numbers that follow specific patterns or rules. An arithmetic progression (AP) is a type of sequence where each term after the first is the result of adding a constant value, called the common difference, to the previous term.
The understanding of AP is crucial in our exercise as it provides a structured approach to balance and equate separate logarithmic expressions. In mathematical terms, if \( a, b, c \) are in AP, then \( 2b = a + c \). This property was utilized in forming our equation in the exercise.

Knowledge of sequences and series allows mathematicians and students alike to solve equations where terms are distributed across different parts of an expression. By recognizing an AP, it becomes possible to set up relationships that simplify problem-solving, as we saw with the logarithmic expressions here. This makes sequences an invaluable tool in the study of mathematical patterns and structures.