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.