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Graph sketching involves plotting the general shape of a graph based on its function and characteristics. To sketch graphs of logarithmic functions like \( y = \log x \) and \( y = \log(\log x) \), we need to understand their behavior.
Logarithmic functions are defined for positive values of \( x \) and display logarithmic growth characteristics. For \( y = \log x \):

• The graph starts at (1,0) since \( \log(1) = 0 \).
• It increases slowly as \( x \) increases, indicating growth.
• There is a vertical asymptote at \( x = 0 \) because \( \log(x) \) approaches negative infinity as \( x \) approaches zero from the right.

For \( y = \log(\log x) \), the graph is defined for \( x > 1 \).
The point (\( e, 0 \)) marks where the graph intersects the x-axis, as \( \log x \) becomes 1. As \( x \) continues to increase, the curve flattens more quickly, indicating slower growth compared to \( y = \log x \).
When sketching, ensure clear demarcation of the domain and characteristic points.

Function analysis helps understand the properties and behavior of mathematical functions. For the logarithmic function \( y = \log x \):

• Domain: x > 0, because the logarithm of zero or a negative number is undefined.
• Range: All real numbers, as the output can be any value from negative infinity to positive infinity.
• Asymptote: Vertical at \( x = 0 \).
• Intercept: The curve crosses the y-axis at (1,0).

Similarly, for \( y = \log(\log x) \), we find:

• Domain: x > 1, because \( \log(x) \) must be greater than zero, requiring \( x > 1 \).
• Range: All real numbers.
• Point of interest: It crosses the x-axis at (\( e \), 0).
• Growth characteristics: The rate of growth is much slower compared to \( y = \log x \).

Use these insights for understanding and explaining the behavior of logarithmic graphs.

Utilizing a graphing utility is an excellent way to visualize functions and verify your manual sketches. Tools like Desmos or graphing calculators are ideal for this. To effectively use a graphing utility for \( y = \log x \) and \( y = \log(\log x) \):
- **Input the Functions:** Enter \( y = \log x \) and \( y = \log(\log x) \) into the utility input field.- **Adjust the View:** Set the domain appropriately, ensuring \( x > 0 \) for \( y = \log x \) and \( x > 1 \) for \( y = \log(\log x) \). Adjust the window settings to highlight these domain restrictions easily.- **Examine Features:** Look for key features such as asymptotes, growth patterns, and intersection points. By comparing the utility's plots with your hand-drawn sketches, you ensure accuracy. These tools offer real-time feedback and adjustment options, enhancing understanding and confidence in graph sketching.