Problem 44 - College Algebra

Question

For the following exercises, sketch the graph of the indicated function. $$f(x)=2 \log (x)$$

Step-by-Step Solution

Answer

The graph is a vertically stretched logarithmic curve with key points at (1,0), (10,2), (100,4) and a vertical asymptote at x=0.

Step 1: Understand the Basic Logarithmic Function

The basic logarithmic function is given by \(f(x) = \log(x)\), which is the logarithm with base 10. It has a domain of \(x > 0\) and a vertical asymptote at \(x = 0\). The graph passes through the point \((1,0)\), since the logarithm of 1 is 0.

Step 2: Apply the Transformation: Stretch Vertically

The given function is \(f(x) = 2 \log(x)\). The factor of 2 causes a vertical stretch of the logarithmic graph. This means that each point \((x, y)\) on the original \(\log(x)\) graph moves to \((x, 2y)\). For example, the point \((10, 1)\) on \(f(x) = \log(x)\) becomes \((10, 2)\) on \(f(x) = 2 \log(x)\).

Step 3: Identify Key Points and Asymptote

Identify important points from the base function and apply the vertical stretch. The key points are: \((1, 0)\), and as logarithms of powers of 10 increase by 1, \((10, 2)\), \((100, 4)\), etc. The vertical asymptote remains at \(x = 0\).

Step 4: Sketch the Graph

Draw the vertical asymptote line at \(x = 0\). Then, plot the key points: \((1, 0)\), \((10, 2)\), \((100, 4)\). Connect these points with a smooth curve that approaches the asymptote as \(x\) approaches 0 from the right and rises slowly as \(x\) increases.

Key Concepts

Vertical Stretch

When dealing with logarithmic functions, one of the key transformations you might encounter is a vertical stretch. This occurs when the original function is multiplied by a constant factor. In the function \(f(x) = 2 \log(x)\), the multiplier is 2, which means every output value of \(\log(x)\) is doubled. To visualize this, consider that each point on the graph of the original function \(\log(x)\) is transformed. For example, if a point on the original graph is \((x, y)\), after the transformation it becomes \((x, 2y)\). The point \((10, 1)\) becomes \((10, 2)\) and \((100, 2)\) becomes \((100, 4)\) after a vertical stretch. A vertical stretch does not affect the position of the asymptote or the x-intercept. However, it makes the graph steeper, reflecting the amplification of the values across the vertical axis.

Logarithmic Transformations

Logarithmic transformations are used to modify the appearance of the graph of a logarithmic function. In the function \(f(x) = 2 \log(x)\), the transformation involves a multiplication that results in a vertical stretch. Transformations are crucial in understanding how functions behave and graphically represent different operations on the logarithm's base function. Here's how to think about them:

• Vertical Stretch: Multiplies all y-values by a factor, making the graph steeper or shallower.
• Horizontal Shifts: Moves the graph left or right. This affects the input values.
• Vertical Shifts: Moves the graph up or down, adding or subtracting a constant to the function.
• Reflections: Flips the graph over the x-axis or y-axis, changing the signs of the coordinates.

Logarithmic transformations help in visualizing changes to the graph due results from modifications to the base function \(\log(x)\), such as changing its steepness or shifting its position on the plane.

Domain of Functions

The domain of a function refers to the set of all possible input values (x-values) that make the function defined. For the logarithmic function \(\log(x)\), the domain is \(x > 0\). This is because you can only take the logarithm of positive numbers.For a function like \(f(x) = 2 \log(x)\), the presence of a vertical stretch does not change the domain from the base function. It remains \(x > 0\) as the logarithm is still only defined for those positive x-values.Understanding the domain is crucial because it tells you where the function can take input from and consequently, where the function graph lies on the x-axis. The logarithmic function will not touch or cross the line \(x = 0\) due to a vertical asymptote here, which is a boundary the graph approaches but never meets.

Key Points in Graph Sketching

When sketching the graph of a logarithmic function, identifying key points is essential for an accurate representation. These points provide a framework that captures the behavior of the graph across its domain.Begin with the base point \((1, 0)\), which is a result of \(\log(1) = 0\). This point remains unchanged even with transformations. In the function \(f(x) = 2 \log(x)\), after applying the vertical stretch, other key points to note include \((10, 2)\) and \((100, 4)\). These points signify how the y-values are being amplified due to the factor of 2 multiplying the logarithmic values.Remember to also consider the vertical asymptote at \(x = 0\). This is a crucial feature that influences how the graph behaves as it approaches this line. Plot these points carefully and draw a smooth curve through them, ensuring the graph approaches the asymptote as x nears zero, rising slowly as x values increase.