Explain how to use the graph of  f(x) = 2x   to graph g(x) = log2 x.   (Use the relationship between the exponential and logarithmic functions.)

To graph \( g(x) = \log_2 x \) using the graph of \( f(x) = 2^x \), you can utilize the relationship between exponential and logarithmic functions. Here’s a step-by-step explanation:

### Relationship Between Exponential and Logarithmic Functions
The functions \( f(x) = 2^x \) and \( g(x) = \log_2 x \) are inverses of each other. This means that each function undoes the action of the other. Specifically:
\[ y = 2^x \quad \text{if and only if} \quad x = \log_2 y \]

### Steps to Use the Graph of \( f(x) = 2^x \) to Graph \( g(x) = \log_2 x \)

1. **Plot \( f(x) = 2^x \)**:
   - Start by plotting the exponential function \( f(x) = 2^x \). 
   - This function passes through the points \((0, 1)\) (since \( 2^0 = 1 \)), \((1, 2)\) (since \( 2^1 = 2 \)), and \((-1, 0.5)\) (since \( 2^{-1} = 0.5 \)).

2. **Reflect Across the Line \( y = x \)**:
   - The graph of \( g(x) = \log_2 x \) can be obtained by reflecting the graph of \( f(x) = 2^x \) across the line \( y = x \).
   - To reflect a point \((a, b)\) on \( f(x) = 2^x \) to the graph of \( g(x) = \log_2 x \), you interchange the coordinates to get \((b, a)\).

3. **Identify Key Points**:
   - The point \((0, 1)\) on \( f(x) = 2^x \) becomes \((1, 0)\) on \( g(x) = \log_2 x \).
   - The point \((1, 2)\) on \( f(x) = 2^x \) becomes \((2, 1)\) on \( g(x) = \log_2 x \).
   - The point \((-1, 0.5)\) on \( f(x) = 2^x \) becomes \((0.5, -1)\) on \( g(x) = \log_2 x \).

4. **Draw the Reflected Graph**:
   - Plot the points \((1, 0)\), \((2, 1)\), and \((0.5, -1)\) on a new set of axes.
   - Draw a smooth curve passing through these points. This curve is the graph of \( g(x) = \log_2 x \).

5. **Verify the Graph**:
   - Ensure that the graph of \( g(x) = \log_2 x \) passes through the origin and rises to the right, approaching infinity as \( x \) increases.
   - Also, ensure that as \( x \) approaches zero from the right, \( g(x) \) approaches negative infinity.

### Summary

- The graph of \( g(x) = \log_2 x \) is the reflection of the graph of \( f(x) = 2^x \) across the line \( y = x \).
- Points on \( f(x) = 2^x \) such as \((a, b)\) become \((b, a)\) on \( g(x) = \log_2 x \).

### Questions

1. What is the inverse relationship between exponential and logarithmic functions?
2. How do you determine key points on the graph of \( f(x) = 2^x \)?
3. What does the reflection of a graph across the line \( y = x \) represent?
4. How do you verify if the reflected graph is accurate?
5. What is the behavior of \( g(x) = \log_2 x \) as \( x \) approaches infinity?
6. What is the behavior of \( g(x) = \log_2 x \) as \( x \) approaches zero?
7. How can you use other exponential functions to graph their corresponding logarithmic functions?
8. What are some real-world applications of exponential and logarithmic functions?

### Tip

When reflecting a graph to find its inverse, always check a few key points to ensure accuracy and confirm the general shape and behavior of the reflected graph.

Learn how to graph the logarithmic function g(x) = log2 x using the exponential function f(x) = 2^x. Understand the inverse relationship between these functions and how to reflect the graph across the line y = x. Suitable for high school students studying functions and their graphs.

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