Solving Logarithmic Equations: A Step-by-Step Guide to Finding x

In the realm of mathematics, particularly in the domain of logarithmic equations, one often encounters situations where specific values of variables must be found through rigorous algebraic manipulations. This article will guide you through the detailed process of solving a pair of logarithmic equations to find the value of x. Specifically, we will explore the equations logqp2 x2 and logpq3 x. Through a series of logical steps, we will arrive at the solution x 31/3. Let's dive into the intricacies.

Understanding the Equations

Our primary challenge is to solve the given equations where logqp x2 (Equation 1) and logpq3 x (Equation 2). A key insight is to recognize the reciprocal relationship between the logarithms. We can manipulate these equations to express them in terms of a single variable.

Manipulating the Equations

Let's start with the first equation:

logqp x2

By definition, this can be rewritten as:

logqp x2 which implies 2

For the second equation, we have:

logpq3 x

Using the properties of logarithms, this can be written as:

3}{ln p} x which simplifies to or 3

Combining the Equations

Now, let's combine these two equations. From the first equation, we have:

2 or ln q 2}

Substituting this into the second equation:

3 becomes 32}}{ln p} x which simplifies to 32} x

Multiplying both sides by x2 gives:

3 x3

Therefore, x 31/3

Conclusion

Through a series of algebraic manipulations, we have demonstrated that the solution to the given logarithmic equations is x 31/3. This process highlights the importance of recognizing and utilizing the properties of logarithms to simplify and solve complex equations. The logical steps we have taken ensure that the solution is both accurate and rigorously derived.