Problem 57 - Precalculus : Building Concepts and Connections

Question

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=2 \log x$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=2 \log x\) is \((0, \infty)\), and the graph of the function has a vertical asymptote at \(x=0\).

Step 1: Finding the domain of the function

The function under consideration is a logarithm, which can only take positive real numbers as input. Therefore, the domain of this function is \((0, \infty)\).

Step 2: Graphing the logarithm function

The graph of \(f(x)=2 \log x\) will be a steeper version of the basic \(y=\log x\) function, because the coefficient of 2 increases the steepness. However, like all logarithmic functions, it will have a vertical asymptote at \(x=0\).

Step 3: Labeling the asymptote

The graph of the function has a vertical asymptote at \(x=0\), this should be clearly labeled on the graph.

Key Concepts

Understanding Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and play a fundamental role in mathematics and various applied sciences. The general form of a logarithmic function is expressed as \( y = \text{log}_b(x) \), where \( b \) is the base and must be a positive real number, different from 1, and \( x \) is the argument of the logarithm, which must also be a positive real number.

For any logarithmic function, the domain is an important aspect as it tells us for which values of \( x \) the function is defined. In the example \( f(x) = 2 \text{log} x \), the base, omitted here, is understood to be 10, a common convention when the base is not explicitly written. Since logarithms can only be taken of positive numbers, the domain of \( f(x) \) is \( (0, \text{infinity}) \). This domain means that logarithmic functions are undefined for zero and negative inputs, which will impact their graphical representations.

Graphing Logarithms

To graph a logarithm function like \( f(x) = 2 \text{log} x \), we start by considering the basic shape of the \( y = \text{log} x \) function. This graph has a characteristic curve that passes through the point \( (1,0) \), rises slowly to the right, and declines steeply to the left, without ever touching the vertical line \( x = 0 \), which is the asymptote.

When we introduce a coefficient, such as 2 in \( f(x) = 2 \text{log} x \), it alters the steepness or the vertical stretch of the graph. The function will still pass through \( (1,0) \) but will rise more quickly as it moves to the right and decline more steeply as it approaches the vertical asymptote. This can be illustrated by plotting a few key points and observing that for every \( x \), the \( y \)-value is twice what it would be in the base \( \text{log} x \) graph.

Key Points to Plot

• \( (1,0) \): The graph will always pass through this point since \( \text{log} 1 = 0 \).
• Points where \( x \) is a power of 10 (like \( (10,1) \), \( (100,2) \)) are helpful because they result in neat integers when considering the base 10 logarithm.
• Points between 0 and 1 (like \( (0.1,-1) \), \( (0.01,-2) \)) help to show the steep decline to the left of the \( y \)-axis.

Asymptotes in Functions

An asymptote is a line that a graph approaches but never actually touches or crosses. In the context of logarithmic functions, vertical asymptotes are particularly important. They indicate points at which the function heads off towards infinity.

For the function \( f(x) = 2 \text{log} x \), the vertical asymptote is at \( x=0 \). This occurs because logarithms are not defined at zero and thus, as \( x \) gets closer to zero from the right, the \( y \)-value of the logarithm function becomes increasingly large in the negative direction. Graphically, this means that the curve gets closer and closer to the line \( x=0 \) but does not reach it.

To properly graph the function with its asymptote, one should draw a dashed vertical line along the asymptote to indicate that the graph will never intersect this line. This representation helps in visualizing the function's behavior near the edges of its domain and in understanding the limits and continuity of logarithmic functions.