Problem 20 - Contemporary Precalculus

Question

Use graphical or algebraic means to determine whether the statement is true or false. $$e^{x \ln x}=x^{x} \quad(x>0) ?$$

Step-by-Step Solution

Answer

Answer: Yes, the equation \(e^{x\ln x}=x^x\) is true for all \(x>0\).

Step 1: Simplify the Left Side of the Equation

To simplify the left side of the equation, we can use the property: \(\text{if } a=b \implies a^c = b^c\). We know that \(a=e^x\) and \(b=x\), we have: $$ e^{x\ln x} = (e^x)^{\ln x} $$

Step 2: Apply the Logarithm Identity

Since \(x^{\ln x}=e^{\ln x^x}\), we can rewrite the left side as: $$ (e^x)^{\ln x} = e^{\ln x^x} $$

Step 3: Simplify the Right Side of the Equation

On the right side, we have \(x^x\). In order to compare it with the left side, we should also express it in the form \(e^{f(x)}\). We can do that by applying the logarithm again as we did on the left side. Since \(x^x=e^{\ln x^x}\), we can rewrite the right side as: $$ x^x = e^{\ln x^x} $$

Step 4: Compare Both Sides

Now both sides of the equation are expressed in the form \(e^{f(x)}\). We have: $$ e^{\ln x^x} = e^{\ln x^x} $$ Since both sides are equal, the statement is true for all \(x > 0\). Therefore, the equation \(e^{x\ln x}=x^x\) is true for \(x>0\).

Key Concepts

Logarithmic Properties

Logarithmic properties are incredibly useful when solving exponential equations. Here, we use the natural logarithm when transforming expressions to compare both sides of the equation. Remember, the natural logarithm, often represented as \(\ln\), is the inverse operation of exponentiation. This means applying \(\ln\) to a function like \(e^x\) can help simplify expressions significantly.

Some key logarithmic properties include:

• \(\ln(a^b) = b \cdot \ln(a)\): This power rule allows us to bring down the exponent, making it easier to work with.
• \(\ln(e^x) = x\): This property reminds us that taking the natural logarithm of \(e\) raised to a power returns the power itself.
• \(e^{\ln(x)} = x\): Often used in simplifying expressions back to their base form when they are written as logarithms.

In the original exercise, we see these properties in action when we express both sides of the equation with \(\ln\) to verify equality.

Exponential Functions

Understanding exponential functions is vital for solving this type of problem. An exponential function has the general form \(f(x) = a^{bx}\), where \(a\) is a positive constant and \(b\) is a real number.

In our context, the function \(e^x\) is a classic example of an exponential function, where \(e\) is the base and \(x\) is the exponent.

It’s also important to recognize that exponential functions grow quickly. In the exercise, the left-hand side of the equation is expressed initially as \(e^{x\ln x}\). By recognizing the structure of exponential functions, we can then compare it to the right-hand side \(x^x\).

Rewriting these expressions using properties of logarithms helps us further unlock the potential solutions by expressing both sides in the potentially simpler base \(e\). This simplification makes the comparison easier.

Algebraic Simplification

Algebraic simplification involves the process of using mathematical operations to make expressions easier to understand or solve. In this exercise, simplification helps to clarify if two expressions are equal, particularly when complex exponentials are involved.

To simplify the equation \(e^{x \ln x} = x^x\), the approach is to transform these expressions using logarithmic identities. You're essentially breaking down the problem into more basic components:

• First, recognize the expression can be rewritten using properties, such as \(e^{a} = a\) when transformed properly with logarithms.
• Express both sides using the same base; in this case, using the natural base \(e\) simplifies comparison.
• Lastly, ensure that each side reaches an identical form, verifying the truthfulness of the initial statement for \(x > 0\).

Through algebraic simplification, complicated exponential expressions become manageable and understandable, revealing the true nature of their relationship.