The question in the image is:

Investigate: Consider the exponential function $y = 2^x$. Write the inverse equation and isolate for the $y$-variable. What problem do you encounter?

To find the inverse of the function $y = 2^x$, let's go through the steps:

1. Rewrite the equation: Start with $y = 2^x$.

2. Swap $x$ and $y$: To find the inverse, interchange $x$ and $y$. This gives: $x = 2^y$

3. Isolate $y$: To solve for $y$, take the logarithm of both sides. Using the natural logarithm $\ln$, we get: $\ln(x) = \ln(2^y)$

4. Apply the power rule for logarithms: This allows us to bring $y$ down: $\ln(x) = y \cdot \ln(2)$

5. Solve for $y$: Divide both sides by $\ln(2)$: $y = \frac{\ln(x)}{\ln(2)}$

Therefore, the inverse function is: $y = \frac{\ln(x)}{\ln(2)}$

Problem Encountered

The problem here is that the inverse function, $y = \frac{\ln(x)}{\ln(2)}$, is only defined for $x > 0$. The original function $y = 2^x$ is defined for all real $x$, but its inverse requires $x$ to be positive, as the logarithmic function $\ln(x)$ is only defined for positive values of $x$.

Would you like more details on this or have any questions?

Here are five related questions to further your understanding:

1. What is the domain of the original function $y = 2^x$?
2. How does the range of $y = 2^x$ relate to the domain of its inverse?
3. What are the key differences between exponential and logarithmic functions?
4. How would the process change if the function were $y = 3^x$ instead of $y = 2^x$?
5. What does the graph of $y = 2^x$ and its inverse look like?

Tip: When finding the inverse of a function, remember to interchange $x$ and $y$ and solve for the new $y$-variable. This helps in reversing the function’s original operation.