Understanding the Equation x^(log x) x^(3/100): A Comprehensive Analysis

Mathematics often presents us with intriguing equations that challenge our understanding of logarithmic and exponential relationships. One such equation is xlog_{10} x xfrac{3}{100}. This article will delve into the nuances of this equation and explore its solutions step by step.

Solving the Equation xlog_{10} x xfrac{3}{100}

Let's start by expressing the equation in a more manageable form. We begin with:

xlog_{10} x xfrac{3}{100}

By taking the logarithm of both sides, we can simplify this equation. Here's a step-by-step breakdown:

We assume u log_{10} x. Substituting u into the equation, we get:

10^{u^2} frac{10^{3u}}{100}

Simplifying the right-hand side:

10^{u^2} frac{10^{3u}}{10^2}

10^{u^2} 10^{3u - 2}

Since the bases are the same, we can equate the exponents:

u^2 3u - 2

Rearranging terms, we get a standard quadratic equation:

u^2 - 3u 2 0

Solving the quadratic equation using the quadratic formula:

u frac{3 pm sqrt{3^2 - 4 times 1 times 2}}{2 times 1}

u frac{3 pm sqrt{9 - 8}}{2}

u frac{3 pm 1}{2}

This gives us two solutions for u:

u 1

u 2

Recall that u log_{10} x. Therefore, we can find the corresponding values of x:

For u 1: x 10^1 10

For u 2: x 10^2 100

Therefore, the solutions to the equation are x 10 and x 100.

Understanding the Components: Logarithm and Exponential

This equation combines logarithmic and exponential functions, making it a powerful tool in advanced mathematics. Let's break down the key components:

1. Logarithm

A logarithm is the inverse operation to exponentiation. In this context, log_{10} x represents the power to which the base 10 must be raised to obtain x. It is a fundamental concept in many areas of mathematics, including calculus, algebra, and number theory.

2. Exponential Function

The exponential function, represented as 10^u, is the inverse of the logarithmic function. Here, 10^u gives us the value of x when u is the power to which 10 is raised.

Applications of the Equation

The equation xlog_{10} x xfrac{3}{100} has practical applications in fields such as:

1. Engineering and Physics

In engineering and physics, such equations are used to model growth rates, decay processes, and other phenomena where exponential and logarithmic relationships are involved. For example, in electrical engineering, these equations can be used to analyze signal strengths and noise levels.

2. Finance and Economics

In finance and economics, similar equations are used to model interest rates, depreciation, and growth rates. Understanding the principles behind these equations can provide insights into how financial markets and economic systems evolve over time.

Conclusion

In conclusion, the equation xlog_{10} x xfrac{3}{100} involves a combination of logarithmic and exponential functions, making it a rich area for mathematical exploration. By solving this equation, we have demonstrated methods for finding solutions using basic algebra and logarithmic identities. Understanding such equations is crucial for advanced mathematical and practical applications in various fields.