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 \( f(x) = 2^x \), we can leverage the relationship between exponential and logarithmic functions. Here's a step-by-step explanation:

1. **Understand the Relationship**:
   - The exponential function \( f(x) = 2^x \) and the logarithmic function \( g(x) = \log_2 x \) are inverses of each other.
   - This means that if \( (a, b) \) is a point on the graph of \( f(x) = 2^x \), then \( (b, a) \) will be a point on the graph of \( g(x) = \log_2 x \).

2. **Plot Points for \( f(x) = 2^x \)**:
   - Generate some points for the function \( f(x) = 2^x \). For example:
     - When \( x = -2 \), \( 2^{-2} = \frac{1}{4} \) ⇒ \( (-2, \frac{1}{4}) \)
     - When \( x = -1 \), \( 2^{-1} = \frac{1}{2} \) ⇒ \( (-1, \frac{1}{2}) \)
     - When \( x = 0 \), \( 2^0 = 1 \) ⇒ \( (0, 1) \)
     - When \( x = 1 \), \( 2^1 = 2 \) ⇒ \( (1, 2) \)
     - When \( x = 2 \), \( 2^2 = 4 \) ⇒ \( (2, 4) \)

3. **Switch Coordinates to Get Points for \( g(x) = \log_2 x \)**:
   - Using the points from \( f(x) = 2^x \), switch the coordinates to get points for \( g(x) = \log_2 x \):
     - \( (-2, \frac{1}{4}) \) becomes \( (\frac{1}{4}, -2) \)
     - \( (-1, \frac{1}{2}) \) becomes \( (\frac{1}{2}, -1) \)
     - \( (0, 1) \) becomes \( (1, 0) \)
     - \( (1, 2) \) becomes \( (2, 1) \)
     - \( (2, 4) \) becomes \( (4, 2) \)

4. **Plot the Points for \( g(x) = \log_2 x \)**:
   - Plot the points \( (\frac{1}{4}, -2) \), \( (\frac{1}{2}, -1) \), \( (1, 0) \), \( (2, 1) \), and \( (4, 2) \) on the coordinate plane.

5. **Draw the Graph**:
   - Connect these points smoothly to form the graph of \( g(x) = \log_2 x \). The graph will pass through these points and should approach the y-axis (but never touch it) as \( x \) approaches 0 from the right, and it will continue to increase slowly as \( x \) increases.

6. **Verify Key Properties**:
   - The graph should pass through the point \( (1, 0) \) since \( \log_2 1 = 0 \).
   - The graph should intersect the x-axis at \( x = 1 \) and have a vertical asymptote at \( x = 0 \).

### Summary of Steps:
1. Plot points for \( f(x) = 2^x \).
2. Switch the coordinates of these points.
3. Plot the new points for \( g(x) = \log_2 x \).
4. Connect the points to form the graph of \( g(x) = \log_2 x \).

### Do you want more details or have any questions?

### Related Questions:
1. How do you graph other logarithmic functions like \( \log_{10} x \)?
2. What are the properties of logarithmic functions?
3. How do you graph \( g(x) = \log_3 x \) using \( f(x) = 3^x \)?
4. How do transformations affect the graph of a logarithmic function?
5. What is the domain and range of \( g(x) = \log_2 x \)?
6. How do you find the inverse of a logarithmic function?
7. How do you solve logarithmic equations?
8. What are common applications of logarithmic functions in real life?

### Tip:
Always remember that the base of the logarithm determines the rate at which the logarithmic function grows. For example, \( \log_2 x \) grows faster than \( \log_{10} x \).

Learn how to graph g(x) = log2 x using the relationship with f(x) = 2^x. This guide provides a detailed step-by-step explanation and plotting process for high school students.

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