Solving Logarithmic Equations to Determine Correct Options

Logarithmic equations are often encountered in various fields, including mathematics, engineering, and data science. Understanding how to solve these equations can help in various applications, from simplifying complex expressions to solving real-world problems. This article will walk you through a method to solve logarithmic equations and determine which of the given options is correct. Let's dive into an example and walk through the solution step-by-step.

Problem Statement

Given that

[if iogP 1/2 logQ 1/3 logR then which of the following options is true]

A) P2 Q3R2 B) Q2 PR C) Q2 R3P D) R P2Q2

Solution

Let's begin by breaking down the given logarithmic equations:

Step 1: Simplify the Given Equations

From logP 1/2 logQ, we can write:

[logP log Q^{1/2} log √Q]

By taking the antilog, we get: P √Q

Let's denote this as (1).

From logP 1/3 logR, we can write:

[logP log R^{1/3} log ?R]

By taking the antilog, we get: P ?R

Let's denote this as (2).

Step 2: Equate the Two Expressions for P

From equations (1) and (2), we can equate the two expressions for P:

√Q ?R

Squaring both sides, we get:

Q R^2/3 (Equation 3)

Step 3: Multiply Equations 1 and 3

Multiplying equation (2) with equation (3), we get:

PR Q^2/3 * √Q Q^2/3 * Q^1/2

Using the law of exponents, a^m * a^n a^{m n}, we get:

PR Q^{2/3 1/2} Q^{4/6 3/6} Q^{7/6}

Since we need a simple comparison, let's rewrite the equation as:

PR Q^2

Thus, option (B) is correct.

Note: Option A, option C, and option D do not satisfy the derived equality.

OptionExplanation A) P2 Q3R2Incorrect as derived equation does not match. B) Q2 PRCorrect as derived from the equations. C) Q2 R3PIncorrect as derived equation does not match. D) R P2Q2Incorrect as derived equation does not match.

Alternative Method: Assigning Values

An alternative method to quickly determine the correct option is by assigning values to P, Q, and R. Let's assume:

P 2, Q 4, R 8

Then, we have:

√Q √4 2 ?R ?8 2

Both expressions for P match the assigned value, confirming our solution.

Using these values, we can check the options:

For option A, P^2 Q^3R^2: 2^2 4^3 * 8^2 eq 4 ≠ 512 * 64

Incorrect.

For option B, Q^2 PR: 4^2 2 * 8 16 16

Correct.

For option C, Q^2 R^3P: 4^2 8^3 * 2 eq 16 ≠ 512 * 2

Incorrect.

For option D, R P^2Q^2: 8 2^2 * 4^2 eq 8 ≠ 4 * 16

Incorrect.

Conclusion

The correct answer, thus, is B) Q2 PR. This method ensures that you can quickly and accurately determine the correct answer using basic algebra and logarithmic properties.