Problem 35 - Algebra 2

Question

Graph each logarithmic function. $$ y=\log _{4} x $$

Step-by-Step Solution

Answer

The graph of \(y=\log_4x\) includes a vertical asymptote at x=0 and an x-interception at x=1. As x tends to infinity, y tends to infinity and as x tends to 0, y tends to negative infinity.

Step 1: Understanding the Parent Function

The logarithmic function given is in the form \(f(x) = \log_bx\). With \(b=4\), the function is \(f(x) = \log_4x\). If we take the exponent on both sides, the equation becomes \(x = 4^y\). This can act as a guide for deriving the table of values.

Step 2: Deriving the Table of Values

Choose a few values of \(y\) then use the formula \(x=4^y\) to calculate \(x\). For instance, when \(y=-1\), \(x=4^{-1}=0.25\); when \(y=0\), \(x=4^0=1\); when \(y=1\), \(x=4^1=4\); when \(y=2\), \(x=4^2=16\).

Step 3: Plotting the Points

On a graph paper, plot these points: (-1,0.25), (0,1), (1,4), (2,16).

Step 4: Drawing the Curve

Using the plotted points, draw a smooth curve. Ensure that the left side of the curve gradually approaches the x-axis but does not touch or cross it because log functions have a vertical asymptote at x = 0.

Step 5: Stating Observations about the Graph

This function has a vertical asymptote at x=0 and an x-interception at x=1. As x tends to infinity, y tends to infinity and as x tends to 0, y tends to negative infinity.

Key Concepts

Graphing Logarithmic Functions

Graphing logarithmic functions is a fundamental skill in understanding how these functions behave visually. To graph a base 4 logarithmic function like \(y = \log_4 x\), one should start by understanding its transformations and points. Begin by noting that the inverse of the logarithm in this case is the exponential function \(x = 4^y\). This relationship helps in determining specific points for plotting.

• Pick some values for \(y\) (e.g., -1, 0, 1, 2).
• Use the inverse function to find corresponding \(x\) values (e.g., when \(y=0\), \(x=1\); when \(y=1\), \(x=4\)).

By plotting these points and smoothly connecting them, you can visualize how the logarithm behaves. The curve should show a steady rise from left to right, without ever touching the vertical line at \(x=0\). This technique is key to mastering the graphing of logarithmic functions.

Logarithmic Function Properties

Logarithmic functions possess several unique properties that distinguish them from other types of functions. Understanding these properties helps to predict the shape and characteristics of their graphs.

• Domain and Range: A logarithmic function like \(y = \log_4 x\) is only defined for positive \(x\)-values, meaning its domain is \(x > 0\). The range, however, can include all real numbers.
• Vertical Asymptote: This function has a vertical asymptote at \(x = 0\), which means the graph will approach but never cross this line.
• X-intercept: Logarithmic functions typically have an x-intercept at \(x=1\) because \(\log_b 1 = 0\) for any base \(b\).

These properties help provide a template for drawing the logs on a graph and determine their interaction with other scientific or mathematical phenomena.

Vertical Asymptotes

Vertical asymptotes in logarithmic functions represent positions where the function value tends toward negative or positive infinity. For the example function, \(y = \log_4 x\), the vertical asymptote occurs at \(x = 0\).

• This asymptote indicates that as \(x\) approaches 0, \(y\) decreases without bound, heading towards \(-\infty\).
• The graph gets closer and closer to the line \(x = 0\) but never touches it.

Understanding vertical asymptotes provides insight into the behavior and limitations of logarithmic functions. This key feature is crucial for accurately interpreting and drawing their curves.

Exponential and Logarithmic Relationship

The relationship between exponential and logarithmic functions is foundational for algebraic manipulation and understanding. Logarithms essentially reverse exponential equations. For a function \(y = \log_4 x\), this means that it can be rewritten in its exponential form, \(x = 4^y\).

• This relationship indicates that the logarithm answers the question 'to what power must the base be raised, to result in a specific value of \(x\)?'
• Every point on a logarithmic graph corresponds to a point on its exponential counterpart, illustrated by swapping the roles of \(x\) and \(y\).
• This duality explains why the logarithmic curve is a reflection across the y-axis of the exponential curve.

This relationship highlights how converting between the two provides a powerful tool for solving problems involving complex algebraic and real-world scenarios.