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The base-10 logarithm, denoted as \(\text{log}_{10} x\), is a logarithmic function where the base is 10. It's often written simply as \(\text{log} x\) in mathematical notation. This function answers the question: What power do we need to raise 10 to get the number x? For example, \(\text{log}_{10}100 = 2\) because 10 raised to the power of 2 is 100. This type of logarithm is useful because our numbering system is base 10, making calculations and interpretations more intuitive in real-world applications like science and engineering.

Key points about the base-10 logarithm:

• It is only defined for positive values of x, meaning \(\text{log}_{10} x\) does not exist for x ≤ 0.
• Its graph passes through the point \((1,0)\) since \(\text{log}_{10} 1 = 0\).
• As \(\text{log}_{10} x\) increases, the rate of increase slows down, which means it's a logarithmic growth—logarithms grow very slowly compared to linear or exponential functions.

Understanding the base-10 logarithm is crucial, especially when graphing or evaluating functions like \(f(x) = x^2 \text{log}_{10} x\) as it influences the behavior of the function significantly.

A graphing calculator is a powerful tool for visualizing mathematical functions. It allows you to enter functions and see their graphs immediately. Follow these steps to graph a function like \(f(x) = x^2 \text{log}_{10} x\) on your graphing calculator:

1. **Turn On and Set Up**: Power on your graphing calculator and navigate to the graphing mode.
2. **Enter the Function**: Input the function using the calculator's buttons. Use the \(\text{log}\) button to insert the base-10 logarithm part of the function. Typically, you would enter this as \(y = x^2 \times\text{log}(x)\).
3. **Adjust the Viewing Window**: Ensure the viewing window appropriately captures the behavior of the function. Since \(\text{log}_{10} x\) is undefined for x ≤ 0, set the domain starting above 0. For example, use an x-range from 0 to 10 and a y-range that showcases the function's growth, like 0 to 100.
4. **Graph the Function**: After inputting the function and adjusting the window, press the graph button. Observe how the graph behaves as x increases. The graphing calculator makes it easy to see trends and make predictions about function behavior.

Using a graphing calculator can simplify the process of understanding complex functions and enhance your ability to interpret their behavior.

Analyzing the behavior of the function \(f(x) = x^2 \text{log}_{10} x\) gives us insights into how the function values change with varying x. Here's a detailed breakdown:

**As \(x\) Approaches Zero**:
When \(x\) is close to zero but positive, \(x^2\) is a very small number, and \(\text{log}_{10}(x)\) is a very large negative number because the logarithm of a number less than 1 is negative. However, the multiplication of a very small positive number by a large negative number approaches zero, meaning \(f(x) \approx 0\).

**As \(x\) Increases**:
Both \(x^2\) and \(\text{log}_{10} x\) increase. However, \(x^2\) increases much more rapidly than \(\text{log}_{10} x\). The quadratic term \(x^2\) causes the function to grow quickly.

• Initially, for small values of \(x\), the increase in \(f(x)\) might seem slow.
• As \(x\) grows larger, \(\text{log}_{10} x\) increases slowly, and \(x^2\) increases very fast, leading to a rapid increase in \(f(x)\).

**Graph Shape**:
The function will have a characteristic shape where it starts small near x=0, dips slightly, and then turns upwards as \(x\) increases. This is due to the combined effects of quadratic growth and logarithmic growth.

Understanding this function's behavior helps in predicting its values and trends. Practice graphing different sections and ranges to see how changes in x affect \(f(x)\).