Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, sketch the graphs of each pair of functions on the same axis. \(f(x)=\log (x)\) and \(g(x)=\log _{\frac{1}{2}}(x)\)

Step-by-Step Solution

Answer

Plot \(f(x) = \log(x)\) increasing, \(g(x) = \log_{\frac{1}{2}}(x)\) decreasing, both through \((1, 0)\), with vertical asymptotes at \(x=0\).

Step 1: Understand the Functions

The functions given are logarithmic functions. The function \(f(x) = \log(x)\) is the common logarithm with base 10 if not specified, while \(g(x) = \log_{\frac{1}{2}}(x)\) is a logarithm with base \(\frac{1}{2}\). The graph of a logarithm with a positive base greater than 1 is increasing, while the graph of a logarithm with a base between 0 and 1 is decreasing.

Step 2: Identify Key Characteristics

For \(f(x) = \log(x)\), the key points include \((1, 0)\) because \(\log(1) = 0\) and there is a vertical asymptote at \(x = 0\). The graph is increasing. For \(g(x) = \log_{\frac{1}{2}}(x)\), we also have \((1, 0)\) because \(\log_{\frac{1}{2}}(1) = 0\), but here the graph decreases and also features a vertical asymptote at \(x = 0\).

Step 3: Sketch the Graph of f(x)

Start by plotting the key points and characteristics: \((1, 0)\) is on the graph. To the left of 1, the graph approaches the vertical asymptote at \(x = 0\) from the left. To the right, the graph continues to rise slowly without bound. Draw a smooth curve that increases slowly and passes through \((1, 0)\).

Step 4: Sketch the Graph of g(x)

Begin with the key point \((1, 0)\) again. For \(x < 1\), the graph dips steeply towards the asymptote \(x = 0\), and for \(x > 1\), it decreases and approaches the x-axis asymptotically from above. Draw a smooth curve passing through \((1, 0)\) that decreases as \(x\) approaches 0 and the positive x-axis.

Step 5: Compare Both Functions' Curves

Both graphs pass through \((1, 0)\). \(f(x)\) rises while \(g(x)\) falls. The important features are the same vertical asymptote at \(x=0\) for both, but opposite monotonic behavior due to different bases.

Key Concepts

Graphing Logarithmic Functions

Graphing a logarithmic function involves understanding its shape, key characteristics, and the domain where it is defined. Logarithmic functions, such as \(f(x) = \log(x)\) and \(g(x) = \log_{\frac{1}{2}}(x)\), are reverse exponential functions. The logarithmic function \(f(x) = \log(x)\) is referred to as the common logarithm, typically implying a base of 10 if not specified. Its graph will feature an increasing curve to the right of the x-axis starting from the vertical asymptote at \(x = 0\).For graphing:

• Identify the key point: \((1, 0)\). At this point, the value of \(x\) makes the logarithm zero because any number to the power of zero equals one.
• Plot the asymptotic behavior: As \(x\) approaches zero from the right, the graph tends towards negative infinity.
• Sketch the general curve: For \(f(x)\), the curve rises gradually without bound to the right of the key point.
• For \(g(x) = \log_{\frac{1}{2}}(x)\), also start at \((1, 0)\) and plot the steep dip to the x-axis as the function decreases.

Spend ample time practicing the sketching, as it will reinforce understanding of gradual and steep incline or decline of logarithmic functions.

Base of a Logarithm

The base of a logarithmic function significantly influences the graph's shape and behavior. The base in a logarithm dictates how the logarithmic value is calculated. For example, in the function \(f(x) = \log(x)\), a base is typically understood to be 10, unless specified differently. Thus, \(\log(10) = 1\), indicating how quickly the log value rises.Base characteristics:

• A base greater than 1 results in an increasing graph because larger successive numbers need to be raised to reach equivalent exponential values.
• A base between 0 and 1, such as in \(g(x) = \log_{\frac{1}{2}}(x)\), leads to a decreasing graph. In this scenario, smaller powers are needed to match equivalent exponential expressions, hence the decline.

Understanding the effect of different bases helps predict the direction of the graph. The base also affects other characteristics like the rate of increase or decrease and the location and behavior around the asymptote.

Asymptotic Behavior

Asymptotic behavior in logarithmic functions reflects how the graph approaches a line or curve without touching it. In both \(f(x) = \log(x)\) and \(g(x) = \log_{\frac{1}{2}}(x)\), there is a vertical asymptote at \(x = 0\). This indicates that the logarithmic functions cannot pass or touch the line \(x = 0\).Key observations:

• At \(x = 0\), the function is undefined, thereby creating an invisible boundary the curves approach infinitely close to but never intersect.
• For positive bases (>1), such as in \(f(x)\), the curve rises gradually as it moves away from the asymptote.
• Conversely, \(g(x)\)'s curve, with a base between 0 and 1, dips towards the asymptote when moving away from \((1, 0)\).

Understanding asymptotes is crucial as it helps predict graph behavior at boundaries of their domain, assisting in achieving accurate and meaningful sketches.