Problem 72 - Algebra A Combined Function

Question

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=\log x $$

Step-by-Step Solution

Answer

Plot points based on \( f(x)=\log x \), such as (0.1, -1), (1, 0), (10, 1), (100, 2), and draw the curve.

Step 1: Understanding the Function

The function given is \( f(x) = \log x \), which represents the logarithm of \( x \). This is the common logarithm with a base of 10. Its domain is \( x > 0 \) since logarithms are not defined for zero or negative numbers.

Step 2: Select Values for \( x \)

Choose a set of values for \( x \) that make sense for the function's domain. Typical values are 0.1, 1, 10, and 100. These will give a good representation of how the function behaves.

Step 3: Evaluate \( f(x) \) for Selected \( x \) Values

Calculate \( f(x) = \log x \) for each selected \( x \):- \( x = 0.1 \), \( f(0.1) = \log(0.1) = -1 \)- \( x = 1 \), \( f(1) = \log(1) = 0 \)- \( x = 10 \), \( f(10) = \log(10) = 1 \)- \( x = 100 \), \( f(100) = \log(100) = 2 \)

Step 4: Plot Ordered Pairs on a Graph

Plot the points (0.1, -1), (1, 0), (10, 1), and (100, 2) on a graph. Make sure to scale the graph appropriately to accommodate the values chosen for \( x \) and \( f(x) \).

Step 5: Draw a Smooth Curve Through the Plotted Points

Draw a smooth curve through the points plotted. The curve should start at the point for \( x = 0.1 \) and gradually increase to the point for \( x = 100 \), reflecting the logarithmic nature of the function.

Key Concepts

Logarithms

A logarithm is a powerful mathematical tool used to solve equations involving exponential growth or decay. Simply put, a logarithm answers the question: "How many times must we multiply a certain number (the base) to get another number?"
For example, in the expression \( \log_{10} 100 \), we are asking, "To what power should we raise 10 to get 100?" The answer is 2, since \( 10^2 = 100 \).
Logarithms have different bases, but the common logarithm, indicated by \( \log \), has a base of 10. This is the type of logarithm used in the function \( f(x) = \log x \).
Key points to remember about logarithms:

• The logarithm of 1 in any base is 0, because any number raised to the power of 0 is 1.
• Logarithms are only defined for positive numbers.
• Logarithmic functions typically have a gradual curve that ascends more slowly as the value of \( x \) increases.

Domain and Range

The concepts of 'domain' and 'range' are essential when we discuss functions. Domain refers to all the possible input values \( x \) that a function can accept without any issues.
In the case of \( f(x) = \log x \), the domain is \( x > 0 \) because logarithms are not defined for zero or negative numbers. If you see this on a graph, you'll notice that it doesn't extend to the left of the y-axis.
The range is the set of all possible output values. For the logarithmic function \( \log x \), the range is all real numbers.
Important takeaways:

• The domain must exclude values that make the function undefined, like non-positive numbers in this case.
• The range of a \( \log x \) function covers all real numbers, indicating its capability to represent any real-world quantity.

Ordered Pairs

Ordered pairs are crucial for plotting functions on a graph. Each ordered pair consists of an \( x \) value and a corresponding function value \( f(x) \), written as \((x, f(x))\).
For the logarithmic function \( f(x) = \log x \), you calculate \( f(x) \) for different \( x \) values to get ordered pairs. These pairs help in visualizing how the function behaves.
For example, consider the ordered pairs from the solution:

• (0.1, -1) comes from \( x = 0.1 \)
• (1, 0) comes from \( x = 1 \)
• (10, 1) comes from \( x = 10 \)
• (100, 2) comes from \( x = 100 \)

These pairs form the backbone of the graph, as they provide a blueprint for drawing the curve. Always choose \( x \) values wisely so they effectively depict the function's behavior.

Function Graphing

Function graphing is how we visually represent a function to understand its behavior. By placing ordered pairs on a graph, you can draw a curve that represents the function.
The graph of a logarithmic function \( f(x) = \log x \) typically has a characteristic shape. It increases slowly and consistently, hugging the y-axis closely before levelling off.
To graph \( f(x) = \log x \), follow these steps:

• Calculate ordered pairs for different \( x \) values that fall within the function's domain.
• Plot these pairs on graph paper or a digital platform, ensuring accurate spacing.
• Draw a smooth curve through the plotted points, making sure it touches each point smoothly.

This curve gives a clear visual representation of how \( f(x) \) behaves across its domain, providing insight into its real-world applications. Remember to pay attention to the scale to accurately reflect trends in data.