https://file.aiquickdraw.com/mathbot/file/1725412072259qyibk968.jpg

The problem in the image involves evaluating the expression \(\left(\frac{b}{a}\right)^{a}\) under the given condition \(a^b = b^a\).

Let's analyze and solve it step by step.

### Given:
\[ a^b = b^a \]

We need to determine the value of the expression:
\[ \left(\frac{b}{a}\right)^a \]

### Step 1: Rewrite the expression
Consider the equation \( a^b = b^a \). Taking the natural logarithm (ln) on both sides:
\[
\ln(a^b) = \ln(b^a)
\]
\[
b \ln(a) = a \ln(b)
\]

### Step 2: Simplify the equation
Dividing both sides by \(ab\):
\[
\frac{\ln(a)}{a} = \frac{\ln(b)}{b}
\]
This indicates that \( a \) and \( b \) are related in such a way that the ratio of their logarithms to their values is equal.

### Step 3: Evaluate \(\left(\frac{b}{a}\right)^a\)
Given the equation \( a^b = b^a \), we know that:

\[
\left(\frac{b}{a}\right)^a = \frac{b^a}{a^a}
\]

Using \( a^b = b^a \), substitute \( b^a = a^b \):
\[
\left(\frac{b}{a}\right)^a = \frac{a^b}{a^a} = a^{b-a}
\]

### Step 4: Final Answer
Thus, the expression simplifies to:
\[
\left(\frac{b}{a}\right)^a = a^{b-a}
\]

Looking at the provided options in the image, the correct answer appears to be \( b \) (option number 4), where the expression simplifies to:
\[
b^{\left(1 - \frac{b}{a}\right)}
\]

### Summary
The correct answer is option **b**.

Would you like more details on any part of the solution or have any other questions? Here are five related questions to consider:

1. How would the solution change if \(a \neq b\) and \(a^b \neq b^a\)?
2. What is the significance of taking logarithms in such equations?
3. Can this problem be solved without using logarithms?
4. How does the equation \(a^b = b^a\) relate to special numbers like \(e\) (Euler's number)?
5. What would the solution look like for specific values of \(a\) and \(b\) (e.g., \(a = 2\), \(b = 4\))?

**Tip:** When dealing with exponential equations involving variables in both the base and exponent, logarithms can often simplify the process and reveal deeper insights into the relationship between the variables.

Given a^b = b^a, what is the value of (b/a)^a?

Explore how to solve the exponential equation (b/a)^a under the condition a^b = b^a. This problem delves into the use of logarithmic properties and exponential equations, making it suitable for students in Grades 10-12.

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