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.