Certainly! To plot the graph of the function f(x) = log3(x) for 0 ≤ x ≤ 10, you can follow these steps:
1. Understand the Function: The base-3 logarithm function increases slowly as x increases, and it is undefined at x = 0. This function will approach negative infinity as x approaches 0 from the right, and it will cross the x-axis at x = 1 since log3(1) = 0.
2. Set the Domain: We're interested in the interval from 0 to 10. However, because logarithms are undefined at 0, we will plot the function for values of x just greater than 0 (e.g., as close to 0 as we can on our graph) to x = 10.
3. Create Values for Plotting: Select a range of x-values within the domain. For a smooth graph, consider many points to accurately show the curve of the function. Typically, you could start just above 0, like 0.01 to avoid the undefined point at 0, and go up to 10, using consistent intervals.
4. Calculate Corresponding y-values: For each x-value in the chosen range, calculate the corresponding y-value using the function. You would convert the base-3 logarithm to a natural logarithm or a base-10 logarithm using a calculator or a software tool that allows this calculation.
5. Plot Points: On a set of axes, begin plotting the calculated points (x, y) on the graph. Because log3(x) will go to negative infinity as x approaches 0 from the right, the plot will come down from the top-left, heading towards the origin but never actually reaching it.
6. Draw the Graph: Connect the points to form the curve of f(x). As you move right on the x-axis, every time you reach a power of 3 (like 1, 3, 9, etc.), the y-value will be an integer (0, 1, 2, etc., respectively).
7. Crossing the Axes: The graph will cross the x-axis at the point (1, 0) since every logarithm function crosses its respective axis at the value of (1, 0).
8. Axes and Labels: Label the x-axis as "x" and the y-axis as "f(x)". Since this graph is a standard logarithmic function, it's also common to mark key points where x is a power of 3.
9. Asymptotic Behavior: Graphically represent the vertical asymptote at x = 0 by a dashed line or by approaching the y-axis without touching or crossing it.
10. Include a Grid and Scale: Depending on your graphing method, a grid can help read the graph more accurately. Make sure the scale is even and properly represents the range of the values for x and y.
After following these steps, you will have a properly plotted graph of f(x) = log3(x) for 0 ≤ x ≤ 10. The curve will start very steep as x approaches zero from the right and gradually flatten out as x gets larger, maintaining a slow increase as x increases.
