Solving the Equation logxy logx logy
The equation logxy logx logy is an intriguing logarithmic identity. This article aims to explore the solution of this equation, providing a clear understanding of the underlying math and the steps involved in reaching the conclusion. Whether you're a student, a teacher, or someone who works with logarithms, this content offers valuable insights and methods for solving such problems.
Understanding the Equation
Let's begin by examining the given equation:
logxy logx logy
Initially, one might be tempted to simplify or directly equate the terms on both sides. However, a careful analysis is required to derive the correct solution.
Step-by-Step Solution
Consider the equation logxy logx logy. Taking the exponential of both sides, we can convert the logarithmic form back to the exponential form:
x^y x^{logx logy}
Let's simplify this further by taking the logarithm on both sides:
y logx logx logy
From the equation y logx logx logy, we can solve for x and y by considering some basic algebraic manipulations. Notice that if we take y logx, the equation simplifies to:
y logx y logx
This is a trivial identity, indicating that our substitution holds true. Therefore, y logx.
Now, substituting y logx back into the original equation, we get:
logx^y logx logy
Since y logx, we substitute and get:
logy^y logx^logx
This simplifies to:
y logy x logx
The equation y logy x logx can be further simplified by considering the properties of exponential functions and logarithms. If x 2 and y 2, the equation holds true:
2 log2 2 log2
This is obviously true, as both sides are equal.
Graphical Interpretation
Graphically, the equation logxy logx logy can be interpreted as a hyperbola. However, since the original equation with logarithms is only valid for positive values of x and y, we should consider only the positive branch of the hyperbola.
A detailed graph of the equation would show the region where the solution lies. For this specific equation, the solution (x, y) (2, 2) is a point on the graph.
Conclusion
The solution to the equation logxy logx logy is (x, y) (2, 2). This is the only pair of values that satisfy the given logarithmic equation. Understanding the properties of logarithms and the substitution method highlighted in this article can help in solving similar equations.