Question

Consider positive real numbers \(x\) and \(y\). Notice that \(4^{3}<3^{4},\) while \(3^{2}>2^{3}\) and \(4^{2}=2^{4} .\) Describe the regions in the first quadrant of the \(x y\) -plane in which \(x^{y}>y^{x}\) and \(x^{y}

Short answer

Answer: 1. For \(x^y > y^x\), the region is above the line \(y = x\), with \(y > 1\). 2. For \(x^y < y^x\), the region is below the line \(y = x\), with \(y < 1\).

Step-by-step solution

Simplify the inequalities using logarithms

First, let's rewrite the inequalities using logarithms. We can do this because logarithms are monotonically increasing functions, which means they preserve the order of values. For any positive real numbers x, y, and a: If \(x > y\), then \(\log_a(x) > \log_a(y)\). If \(x < y\), then \(\log_a(x) < \log_a(y)\). For the given inequalities, take the logarithm of both sides with base x: 1. For \(x^y > y^x\), we have \(\log_x(x^y) > \log_x(y^x)\). 2. For \(x^y < y^x\), we have \(\log_x(x^y) < \log_x(y^x)\). Now, use the property of logarithms that states \(\log_a(b^c) = c\log_a(b)\): 1. For \(x^y > y^x\), we have \(y\log_x(x) > x\log_x(y)\). 2. For \(x^y < y^x\), we have \(y\log_x(x) < x\log_x(y)\).

Simplify the inequalities further

Next, let's simplify these inequalities further. Notice that for any positive real number x, \(\log_x(x) = 1\). So, we can rewrite the inequalities as follows: 1. For \(x^y > y^x\), we have \(y > x\log_x(y)\). 2. For \(x^y < y^x\), we have \(y < x\log_x(y)\).

Find the regions satisfying each inequality

Now, we need to investigate the regions in the first quadrant where the inequalities hold: 1. For \(x^y > y^x\): In this case, we need to find the region where y is greater than \(x\log_x(y)\). Note that when \(x = y\), the inequality simplifies to \(y > y\log_y(y)\), which implies \(1 > \log_y(y)\). This will be true for all \(y > 1\). Meaning, for values of \(x < y\) and \(y > 1\), the inequality \(x^y > y^x\) holds. The region is, therefore, above the line \(y = x\), with \(y > 1\). 2. For \(x^y < y^x\): In this case, we need to find the region where y is less than \(x\log_x(y)\). As discussed earlier, when \(x = y\), the inequality simplifies to \(1 > \log_y(y)\). This will be true for all \(y < 1\). Meaning, for values of \(x > y\) and \(y < 1\), the inequality \(x^y < y^x\) holds. The region is, therefore, below the line \(y = x\), with \(y < 1\). In conclusion, the regions in the first quadrant of the xy-plane where the given inequalities hold are as follows: 1. For \(x^y > y^x\), the region is above the line \(y = x\), with \(y > 1\). 2. For \(x^y < y^x\), the region is below the line \(y = x\), with \(y < 1\).

Key concepts

Logarithmic Properties

Logarithms are a powerful mathematical tool that relate the exponents of numbers to their values. They allow us to simplify complex exponential inequalities by converting them into more manageable forms. This is achieved using a few important logarithmic properties, such as the fact that logarithms are monotonically increasing functions—meaning the order of numbers is preserved when we take their logarithms.

• If a number is larger than another, its logarithm will also be larger.
• This property helps in handling inequalities, like deciding if one power of a number is greater than another power of a different number.

We use the property \[\log_a(b^c) = c \cdot \log_a(b)\]to transform exponential expressions, making them easier to work with. For example, when we have an inequality of the form \(x^y > y^x\), we can take logarithms on both sides to arrive at \(y > x \cdot \log_x(y)\). Similarly, for \(x^y < y^x\), it becomes \(y < x \cdot \log_x(y)\). These transformations provide clear, comparable terms that can guide us in exploring solutions within specific regions or boundaries, especially when dealing with inequalities within variable constraints.

First Quadrant Regions

The concept of exploring regions, particularly in the first quadrant of the Cartesian plane, is central when describing where certain inequalities hold true. The first quadrant is the section where both coordinates, \(x\) and \(y\), are positive. This is applicable in many mathematical explorations that involve exponential and logarithmic functions.

For exponential inequalities such as \(x^y > y^x\) and \(x^y < y^x\), identifying the regions in the first quadrant where these inequalities are satisfied allows us to graphically visualize and solve these problems. In our current context:

• For \(x^y > y^x\), the region of interest is above the line \(y = x\) and specifically where \(y > 1\).
• For \(x^y < y^x\), it lies below the line \(y = x\) and within the area where \(y < 1\).

These positions suggest a comparison of relative values and a grasp of where these mathematical conditions are true. It's like drawing invisible lines that guide us on where these mathematical relationships hold, demonstrating how one inequality is satisfied under different conditions of \(x\) and \(y\).

Transcendental Functions

Transcendental functions are a type of function that are not algebraic, meaning they cannot be expressed as the root of any polynomial equation with rational coefficients. Common transcendental functions include exponential functions, logarithms, and trigonometric functions.

• These functions have peculiar characteristics and extend beyond regular algebraic functions in terms of complexity and behavior.
• In the case of exponential functions, they often involve irrational or non-repeating growth, which is part of what makes them transcendental.

Exponential expressions such as \(x^y\) and \(y^x\) are transcendental because they incorporate powers that are not fixed, often subject to the behavior of other variables, represented by expressions like \(a^b\) where at least one of \(a\) or \(b\) is a non-integer.

These expressions display interesting dynamic behavior, particularly when we compare them as in our inequalities.
Understanding transcendental functions aids in making sense of why the expression \(x^y\) can be greater than, less than, or equal to \(y^x\) under different conditions and constraints involving \(x\) and \(y\). The unpredictability and non-linear nature of transcendental functions is exactly what makes them so useful for modeling real-world phenomena and complex mathematical ideas.