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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?

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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.

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

Learn how to find the inverse of the exponential function y = 2^x, isolate the y-variable, and understand the challenges encountered with domain restrictions. This problem covers key concepts in exponential functions, logarithms, and inverse functions. Suitable for high school students in Grades 10-12.

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