Finding the Solution for the Equation 3x 5x 7x 150 Using Newton’s Method

In this article, we will explore the solution of the equation 3x 5x 7x 150. This involves identifying the value of x that satisfies the given equation. We will use mathematical analysis and numerical methods to find the solution with high precision.

Introduction to the Equation

The given equation is:

3x 5x 7x - 150 0

This is a transcendental equation, which has no closed-form solution. Therefore, we will employ numeric methods to approximate the root of the equation. We start by analyzing the behavior of the function through plotting and derivative analysis.

Graphical and Differential Analysis

Graphing the function f(x) 3x 5x 7x - 150, we observe a single peak, indicating the function being strictly monotonically increasing. The derivative of the function is given by:

F'(x) 7xln(7) 5xln(5) 3xln(3)

Since the logarithmic terms are positive and the base values are greater than 1, the derivative is positive for all real values of x, confirming the function is strictly monotonically increasing.

Given that f(2) 0 and f(3) 0, we can conclude that the function has exactly one real root within the range 2 x 3. Using Newton's method, we can approximate this root to high precision.

Newton’s Method Application

Newton's method is an iterative process used to find successively better approximations to the roots (or zeroes) of a real-valued function. The formula for Newton's method is:

xn 1 xn - f(xn) / f'(xn)

Starting with an initial guess of x 2.35, we apply the method to iteratively improve our approximation:

f(x) 3x 5x 7x - 150 F'(x) ln(3)3x ln(5)5x ln(7)7x

By iterating the formula, we find that the root converges to:

x ≈ 2.335351512 to 10 decimal places.

Complex Domain Consideration

It is important to note that in the complex domain, there can be an infinite number of roots. For a detailed discussion on finding these complex roots, refer to the similar question answered by Jonathan Devors:

Jonathan Devors answer to What is x if 3x 7x 370

Understanding the behavior of the function in both real and complex domains is crucial for a comprehensive mathematical analysis.

Conclusion

In this article, we have explored the solution to the equation 3x 5x 7x 150 using mathematical analysis and numerical methods. By leveraging Newton's method, we were able to approximate the root to high precision. The process involves understanding the behavior of the function through both graphical and differential analysis, and iteratively refining the approximation using numerical techniques.

To further explore the topic and delve into more complex mathematical concepts, refer to the additional resources and discussions available on the web and in relevant literature.

Additional Resources

Find complex roots of similar equations Further reading on numerical methods Explore differential equations and their applications