Problem 7 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\log _{\sqrt{5}} x+\log _{5^{n}} x+\log _{5^{n}} x+\ldots\) upto 7 terms \(=35\), then \(x\) is equal to (A) 5 (B) 25 (C) 125 (D) None of these

Step-by-Step Solution

Answer

None of these (not sufficient information for a clear value).

Step 1: Write the given series

The series given in the exercise is \( \log _{\sqrt{5}} x + 6 \cdot \log _{5^{n}} x \). Here the first term is from the base \( \sqrt{5} \) and the next six terms are based on \( 5^n \).

Step 2: Apply logarithm base change formula

We know the logarithm base change formula: \( \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \). So, \( \log _{\sqrt{5}} x = \frac{\log x}{\log \sqrt{5}} = \frac{\log x}{\frac{1}{2} \log 5} = \frac{2 \log x}{\log 5} \).

Step 3: Simplify the second part of the series

Each term in \( \log _{5^n} x \) can be simplified using the base change formula \( \log _{5^n} x = \frac{\log x}{n \log 5} \).

Step 4: Sum the series terms

By combining the terms of the series, we have: \( \frac{2 \log x}{\log 5} + 6 \cdot \frac{\log x}{n \log 5} = 35 \).

Step 5: Factor out common terms

Factor \( \log x \) out of the equation resulting in \( \log x \left( \frac{2}{\log 5} + \frac{6}{n \log 5} \right) = 35 \).

Step 6: Simplify the equation

This simplifies further to \( \log x \left( \frac{2n + 6}{n \log 5} \right) = 35 \). Since there are no integer n values given in the problem, infer that: \( \log x = 35 \cdot \frac{n \log 5}{2n + 6} \).

Step 7: Solve for x

Upon further inspection of possible values that satisfy such an equation typically involves the assumption of common base values. Since \( \log x = 35 \) typically indicates a clear power related to a base (considering common logarithm usage or decimal), one concludes by trying simple resolution methods or matching standard answers.

Key Concepts

Logarithm Base Change Formula

The logarithm base change formula is an essential tool in performing calculations with different bases. This formula allows you to transform a logarithm with any base into a logarithm of another base of your choice. The formula can be expressed as:
\[\log_{a} b = \frac{\log_{c} b}{\log_{c} a}\]
This expression is particularly helpful when a base is uncommon or makes calculations challenging. You can choose a base, such as 10 or \( e \), for which you have a basic calculator function.

• For instance, transforming \( \log_{\sqrt{5}} x \) into \( \log x / \frac{1}{2} \log 5 \) uses this formula. This conversion makes solving equations easier because standard logarithm tables or calculators often do not support root bases directly.
• Another example is \( \log_{5^n} x \), which simplifies using the same principle into \( \log x / n \log 5 \). By using the formula, you avoid complex or cumbersome manual calculations.

Logarithmic Equation

A logarithmic equation involves logarithms of an unknown quantity, usually set equal to a number. Solving these equations often requires using properties of logarithms to simplify or manipulate the expressions.
Understanding how to manipulate these equations is crucial for solving them. The key steps typically include:

• Applying the base change formula if necessary, to work with bases you can calculate.
• Summing or simplifying logarithmic terms when they appear in series.
• Rearranging the equation to isolate the logarithmic term, which can involve factoring out common terms, as seen in this exercise.
• Finally, exponentiating to solve for the unknown, often by making both sides of the equation powers of the same base.

In the current exercise, we are given a series of logarithmic terms equated to a number. Factoring out \( \log x \) helps simplify the equation, after which typical solving involves testing values for \( x \) or applying numerical methods.

Mathematical Series

A mathematical series is the sum of the terms of a sequence. In this context, the logarithmic series described involves a sum of logarithmic expressions. Understanding the series' nature can greatly aid in solving the problem.
Characteristics of a mathematical series include:

• The number of terms being summed, as each term has specific properties that contribute to the whole.
• Whether the terms exhibit a pattern or follow a rule that makes them easier to handle collectively.
• Evaluating the series usually involves recognizing such patterns or consolidating similar terms to facilitate solving for a single unknown.

This exercise involves summing 7 logarithmic terms, where initial recognition of a pattern or simplifying factor (like a factor of \( \log x \) in each term) helps condense the problem. By integrating such insights into your problem-solving strategy, you often convert a complex series into a manageable equation.