Question

If mean of \(\log x, \log 2 x, \log 8, \log 4 x, \log 4, \log x\) is \(\log 4\) then \(x=\) (where \(x>0)\) (a) 4 (b) 2 (c) 8 (d) 16

Short answer

The value of \(x\) that satisfies the given condition is \(x = 4\).

Step-by-step solution

Write down the given information

The mean of the given expressions is \(\log 4\). So, the sum of these expressions divided by the number of terms (6) is equal to \(\log 4\). The expressions are: 1. \(\log x\) 2. \(\log(2x)\) 3. \(\log 8\) 4. \(\log(4x)\) 5. \(\log 4\) 6. \(\log x\)

Calculate the sum of the expressions

Add all the given expressions: \(\log x + \log(2x) + \log 8 + \log(4x) + \log 4 +\log x\) By the properties of logarithms, we can combine these: \(\log(x^2) + \log(8x^2) + \log 32\)

Use the Mean formula

Now, we will use the formula for the mean: sum of terms / number of terms Sum of terms: \(\log(x^2) + \log(8x^2) + \log 32\) Number of terms: 6 Mean: \(\log 4\) \(\frac{\log(x^2) + \log(8x^2) + \log 32}{6} = \log 4\)

Solve for x

To solve for x, we can first multiply both sides of the equation by 6: \(\log(x^2) + \log(8x^2) + \log 32 = 6\log 4\) Now, we can combine the logarithmic terms on the left-hand side: \(\log(8x^4 \cdot 32) = 6\log 4\) \(\log(256x^4)= 6\log 4\) For this equation to be true, the argument of both logarithms must be equal: \(256x^4 = 4^6\) We can rewrite \(4^6\) as \(256^2\). So, \(256x^4 = 256^2\) Now, divide both sides by 256: \(x^4 = 256\) Now, take the fourth root of both sides: \(x = 256^{\frac{1}{4}}\) \(x = 4\) So, the correct option is: (a) 4

Key concepts

Logarithmic Properties

Understanding logarithmic properties is essential for solving equations that involve logarithms. A logarithm, represented as \(\log_b a\), is the power to which a base \(b\) must be raised to obtain the number \(a\). For instance, \(\log_2 8 = 3\) because \(2^3 = 8\).

Key properties include:

• Product Rule: \(\log_b (MN) = \log_b M + \log_b N\)
• Quotient Rule: \(\log_b (M/N) = \log_b M - \log_b N\)
• Power Rule: \(\log_b (M^k) = k \cdot \log_b M\)
• Change of Base Formula: \(\log_b a = \frac{\log_c a}{\log_c b}\), where \(c\) is any positive value.
• Logarithm Base Switching: \(\log_b a = \frac{1}{\log_a b}\).

These properties are not just academic; they're the tools required to simplify and solve logarithmic equations efficiently, as demonstrated in the provided problem solution. By understanding and applying these properties, students can transform complex logarithmic expressions, making it easier to find the unknown variable.

Mean of Logarithms

The mean (average) of numbers is a concept that can also be applied to logarithms. To find the mean of logarithmic expressions, you sum up the individual logarithms and then divide by the number of terms.

For example, if you have a set of logarithms \(\log b(a_1), \log b(a_2), ..., \log b(a_n)\), their mean would be:
\[\frac{1}{n}(\log b(a_1) + \log b(a_2) + ... + \log b(a_n))\]
This can often be simplified using logarithmic properties. When the logarithms share a common base, and their arguments can be multiplied or divided, the mean can sometimes be expressed as a single logarithm. This concept was applied in the exercise where the mean of six logarithmic expressions was equated to \(\log 4\), allowing for the solution of the variable \(x\).

Solving Logarithmic Equations

Solving logarithmic equations involves isolating the logarithmic term and using properties of logarithms to solve for the unknown. The general steps include:

1. Consolidating all logarithmic expressions on one side of the equation.
2. Using logarithmic properties to combine them into a single logarithmic expression, if possible.
3. Eliminating the logarithm by exponentiating, ensuring that the base of the logarithm becomes the base of the exponent on the other side of the equation.
4. Solving the remaining algebraic equation for the unknown variable.

Note that when solving logarithmic equations, it’s crucial to check that the solutions make sense in the context of the problem since logarithms are only defined for positive real numbers. Therefore, any solution that results in a logarithm of a non-positive number must be discarded. As shown in the provided solution, after simplifying using logarithm properties, the resulting algebraic equation can be solved through standard methods such as taking roots.

JEE Logarithm Problems

Logarithm problems are a staple in JEE Maths due to their complexity and the deep understanding of algebra they require. The Joint Entrance Examination (JEE) is highly competitive, and logarithmic questions often test the ability to apply various mathematical concepts together.

To excel in JEE logarithm problems:

• Practice the application of logarithmic properties extensively.
• Understand how mean (average) applies to logarithmic functions and what it represents.
• Develop a robust process for solving logarithmic equations — often combining algebraic manipulation skills with a conceptual understanding of logarithms.
• Learn to quickly identify when a logarithmic equation has no solution or multiple solutions based on the domain of the function.

By mastering these skills, students can solve complex logarithmic equations like the one presented in the exercise, thereby enhancing their problem-solving abilities for the JEE Maths section.