Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(f(x)=\log _{2} x\)

Step-by-Step Solution

Answer

The function \(f(x)=\log _{2} x\) has a domain of \(x > 0\), a vertical asymptote at \(x = 0\), and an x-intercept at \(x=1\). Its graph starts from the y-axis (the vertical asymptote) and increases to the right, crossing the x-axis at \(1, 0\).

Step 1: Find the Domain

Domain would be the set of all real values for which the function is defined. In this case, the function \(f(x) = \log_2x\) is only undefined for values of \(x\) less than or equal to 0. Therefore, the domain of \(f(x) = \log_2x\) is \(x > 0\).

Step 2: Find the Vertical Asymptote

A vertical asymptote of a function is a vertical line \(x = a\) where the function approaches infinity (either positive or negative ) as \(x\) approaches \(a\). In this case, as \(x\) approaches 0 from the right, \(\log_2x\) goes to -∞, so \(x = 0\) is the vertical asymptote.

Step 3: Find the x-Intercept

The x-intercept of a function is a point where the function crosses the x-axis. In other words, it's the x-value where the function equals 0. By setting our function \(f(x)=\log_2x\) to 0, we can find the x-intercept. So, 0=\(\log_2x\). To solve for \(x\), we can convert the equation into exponential form: \(x=2^0 = 1\). So the x-intercept of our function is \(x=1\).

Step 4: Sketch the Graph

Given the identified x-intercept and vertical asymptote, the function \(f(x) = \log_2x\) can be graphed. Mark the x-intercept at \(x=1\) by plotting a point at \(1,0\). Draw a vertical line \(x=0\) to represent the vertical asymptote. The graph of the function starts from the asymptote and increases as x goes to ∞.

Key Concepts

Domain of a Logarithmic Function

When we're finding the domain of a logarithmic function, like \( f(x) = \log_2 x \), we need to identify the set of all possible \( x \)-values where the function is actually defined.

Logarithms are not defined for zero or negative numbers. That means for \( f(x) = \log_2 x \), \( x \) must be greater than 0, because these are the only values for which the logarithm will produce a real number.

Hence, the domain of \( f(x) = \log_2 x \) is often expressed as:
- \( x > 0 \).

Understanding this concept across any logarithmic function is crucial:

• The domain is restricted to positive real numbers.
• Logarithmic functions are undefined for non-positive values of \( x \).
• This information helps us identify parts of the graph that aren't drawn.

Understanding Vertical Asymptotes

A vertical asymptote is essentially an invisible boundary that the graph of a function "hugs" very closely but never actually touches or crosses. For \( f(x) = \log_2 x \), the graph has a vertical asymptote at \( x = 0 \).

Why is this? As \( x \) approaches zero from the positive side, \( \log_2 x \) heads towards negative infinity, meaning the function values decrease indefinitely but never reach zero.

Here are some helpful points about vertical asymptotes in logarithmic functions:

• They occur where the argument of the logarithmic function (e.g., \( x \) in \( \log_2 x \)) would be zero.
• They're not actual values on the graph but rather indicate a behavior of the function as it tends towards them.
• Always mark it as a dashed vertical line while sketching along the \( x \) value that can never be reached.

These asymptotes help guide the shape and direction of logarithmic graphs.

Finding the x-Intercept

Finding the \( x \)-intercept of a logarithmic function provides the point where the graph crosses the \( x \)-axis. This occurs where the function's value is zero.

For the function \( f(x) = \log_2 x \), we set the function equal to zero: \( \log_2 x = 0 \). Solving for \( x \), the equation converts to an exponent:

\[ x = 2^0 = 1 \]

Thus, the \( x \)-intercept is at the point \( (1, 0) \). This means that when plotting the graph, it will pass through \( (1,0) \).

Some important points about \( x \)-intercepts in logarithmic functions include:

• To find the \( x \)-intercept, solve \( \log_b x = 0 \) which simplifies to \( x = b^0 = 1 \).
• The \( x \)-intercept is always the base raised to the power of zero, so it is 1 for any logarithmic function \( \log_b x \) where \( b \) is the base.
• This point helps in starting the graph and determining its course from thereon.

Recognizing the \( x \)-intercept helps in accurately graphing the behavior of the function.