Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Graph each function. $$f(x)=\log _{5} x$$

Step-by-Step Solution

Answer

The graph of \( f(x) = \log_5 x \) passes through points like \( (1, 0) \), \( (5, 1) \), and has a vertical asymptote at \( x = 0 \).

Step 1: Understand the Logarithmic Function

The function given is \( f(x) = \log_5 x \). This is a logarithmic function with base 5. Logarithmic functions are the inverse of exponential functions, and \( \log_5 x \) is the power to which 5 must be raised to obtain \( x \).

Step 2: Identify the Domain and Range

The domain of \( f(x) = \log_5 x \) is \((0, \infty)\) because you cannot take the logarithm of a non-positive number. The range of this function is \((-\infty, \infty)\).

Step 3: Determine Key Points

To plot the graph, identify key points: \( (1, 0) \) since \( \log_5 1 = 0 \), \( (5, 1) \) since \( \log_5 5 = 1 \), and \( (25, 2) \) since \( \log_5 25 = 2 \). Also, note that as \( x \to 0^+ \), \( f(x) \to -\infty \).

Step 4: Plot the Points on the Graph

Plot the identified points on a coordinate plane: \( (1, 0) \), \( (5, 1) \), and \( (25, 2) \). These points help in forming the shape of the graph.

Step 5: Draw the Curve

Draw a smooth curve through the points \( (1, 0) \), \( (5, 1) \), and \( (25, 2) \). Make sure the curve approaches the y-axis but never touches it as \( x \to 0^+ \), representing the vertical asymptote at \( x = 0 \). The curve continues to rise steadily as \( x \) increases.

Key Concepts

Inverse Functions

Inverse functions reverse the operations of original functions. Here's why understanding inverse functions is crucial for logarithms:

• In a logarithmic function like \( f(x) = \log_5 x \), the inverse is an exponential function \( g(x) = 5^x \).
• Exponential functions are all about repeated multiplication, while logarithmic functions find the power needed to produce a given number.
• Think of \( \log_5 x \) as asking "5 raised to what power gives me \( x \)?"

Knowing this connection helps in graphing and solving equations that involve either of these function types. It’s like having a map that links the two mathematical worlds. Each point on a logarithmic graph corresponds to a reflection over the line \( y = x \) on its exponential counterpart's graph.
As you study, remember this inverse relationship simplifies understanding how inputs and outputs of one function relate to the other.

Domain and Range

Every function has a set of possible inputs (domain) and a set of possible outputs (range). Here’s what that means specifically for logarithmic functions:

• For \( f(x) = \log_5 x \), the domain consists of all positive numbers: \((0, \infty)\). This means you can only take the log of numbers greater than zero.
• The range, however, is \((-\infty, \infty)\). Logarithmic functions can produce any real number as an output, depending on the input value.

This domain is bounded on the left, meaning it doesn’t include zero or negative numbers. Understanding these sets is fundamental when considering where and how your graph appears on a coordinate plane. The unrestricted range reveals that as long as the input is positive, the function can output any real number. Graphically, this is demonstrated by how the curve of the function moves endlessly upwards and downwards along the y-axis.
Knowing the domain helps us avoid undefined scenarios, while the range tells us all possible outcomes of the function.

Graphing Techniques

Graphing a logarithmic function involves certain steps that ensure we capture the essence of the function accurately. Let's explore the key techniques:

• Identify Key Points: These are specific points that are simple to compute and help shape the overall graph. Examples for \( \log_5 x \) include \((1, 0)\), \((5, 1)\), and \((25, 2)\).
• Understand Behavior: As \( x \) approaches 0 from the positive side, \( f(x) \) heads towards negative infinity, indicating a steep descent near the y-axis.
• Smooth Curve Drawing: Connect the points with a smooth curve ensuring it follows the function's continuous progression from left to right.
• Steady Rise: Ensure the curve steadily rises as \( x \) increases, demonstrating the increase in power to achieve higher values of \( x \) in the log equation.

These techniques, like plotting key points and recognizing asymptotic behavior, are foundational in graphing. They help visualize abstract mathematical concepts, aiding further comprehension.

Vertical Asymptotes

Logarithmic functions often exhibit a key behavior near certain values known as vertical asymptotes. Here’s a deeper look into this concept:

• A vertical asymptote is a vertical line that the graph of a function approaches but never touches or crosses.
• For \( f(x) = \log_5 x \), the graph has a vertical asymptote at \( x = 0 \). This tells us that as \( x \) gets closer to zero from the positive side, the output of the function heads toward negative infinity.

In practical terms, vertical asymptotes signify restrictions in the domain of a function. They act as invisible boundaries that the function cannot reach or pass through. Understanding this helps us make accurate predictions and know the behavior of the logarithmic function around these critical points. As you practice, visualize this asymptote as a type of gravitational boundary: no matter how close the graph gets, it never actually touches that line.
By recognizing vertical asymptotes, you'll gain deeper insights into the behavior and limitations of logarithmic and other types of functions.