Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, sketch the graphs of each pair of functions on the same axis. \(f(x)=\log _{4}(x)\) and \(g(x)=\ln (x)\)

Step-by-Step Solution

Answer

Sketch the graphs of \(f(x) = \log_{4}(x)\) and \(g(x) = \ln(x)\) both passing through (1,0) with \(g(x)\) rising faster past \(x=1\).

Step 1: Understand the functions

We are given two functions: \(f(x) = \log_{4}(x)\) and \(g(x) = \ln(x)\). The function \(\log_{4}(x)\) is the logarithm base 4 of \(x\), and \(\ln(x)\) is the natural logarithm (logarithm base \(e\)) of \(x\). Both functions are only defined for \(x > 0\).

Step 2: Determine the domain

Both functions, \(f(x) = \log_{4}(x)\) and \(g(x) = \ln(x)\), are defined for all \(x > 0\). Therefore, the domain for both functions is \((0, \infty)\). This means we will only consider positive values for \(x\) when sketching the graphs.

Step 3: Identify key points for each function

For \(f(x) = \log_{4}(x)\):- At \(x = 1\), \(f(1) = \log_{4}(1) = 0\).- At \(x = 4\), \(f(4) = \log_{4}(4) = 1\).For \(g(x) = \ln(x)\):- At \(x = 1\), \(g(1) = \ln(1) = 0\).- At \(x = e\) (approximately 2.718), \(g(e) = \ln(e) = 1\).

Step 4: Sketch the graph of \(f(x) = \log_{4}(x)\)

Plot the key points: (1,0) and (4,1). The graph of \(f(x) = \log_{4}(x)\) passes through these points and approaches the vertical axis \(y\)-axis asymptotically. It increases slowly beyond \(x = 4\).

Step 5: Sketch the graph of \(g(x) = \ln(x)\)

Plot the key points: (1,0) and roughly close to \(x=2.718\), where \(g(x)\) would be at (e,1). The graph of \(g(x) = \ln(x)\) also passes through these points, approaches the \(y\)-axis asymptotically, and increases more rapidly than \(f(x)\) as \(x\) becomes larger.

Step 6: Compare the graphs

Plot both functions on the same axis. Note that \(g(x) = \ln(x)\) grows faster than \(f(x) = \log_{4}(x)\) as \(x\) increases. Both graphs pass through the point (1,0) and approach the y-axis asymptotically from the right.

Key Concepts

Domain of a Function

In mathematics, understanding the domain of a function is crucial. The domain refers to all possible input values (usually represented by "x") for which the function is defined and produces real numbers.
For logarithmic functions like \(f(x) = \log_{4}(x)\) and \(g(x) = \ln(x)\), the domain is determined by the condition that the argument of the logarithm must be positive. This means both functions are only defined for values of \(x\) that are greater than zero.
Thus, the domain for both \(f(x)\) and \(g(x)\) is the interval \((0, \infty)\), indicating that only positive numbers can be plugged into the function.
This is vital to remember when sketching the graphs or solving problems involving logarithmic functions. Without taking the domain into consideration, you might graph or use the function improperly, leading to incorrect results.

Asymptotic Behavior

Asymptotic behavior in functions describes how the function behaves as it approaches a certain value, typically towards infinity or zero. For the functions \(f(x) = \log_{4}(x)\) and \(g(x) = \ln(x)\), both exhibit asymptotic behavior as they approach the y-axis from the right-hand side.
An asymptote is a line that a graph approaches but never touches or intersects. With logarithmic functions like these, the y-axis (or \(x=0\)) serves as a vertical asymptote.
This indicates that as \(x\) approaches zero from the positive side, the value of the function decreases towards negative infinity, while as \(x\) becomes very large, the function increases but at a gradually decreasing rate.
Understanding asymptotic behavior is important in graphing because it helps in predicting and describing the direction and extent of the graph without computing innumerable points.

Graphing Functions

Graphing functions, especially logarithmic ones, is a visual way to understand their behavior and relationships. Both \(f(x) = \log_{4}(x)\) and \(g(x) = \ln(x)\) are logarithmic functions but with different bases: 4 and \(e\), respectively.
When graphing, it is essential to identify key points. For both functions, a crucial point is \(x=1\), where they intersect the x-axis (both having value 0). Additionally, points like \((4,1)\) for \(\log_{4}(x)\) and \((e,1)\) for \(\ln(x)\) help define their curves.
Use these key points to draw the curve, keeping in mind that both functions will approach the y-axis asymptotically while rising slowly to the right. \(\ln(x)\), being the natural logarithm with the base \(e\), increases more rapidly compared to \(\log_{4}(x)\).
Graphing enables comparison and illustration of the differences in growth rates and characteristics of various logarithmic functions.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a logarithm with base \(e\), where \(e\) is an irrational number approximately equal to 2.718. The natural logarithm is unique because it often appears in mathematical models involving exponential growth or decay.
In the context of the exercise with \(g(x) = \ln(x)\), it offers a faster growth rate compared to logarithms of other bases.
Understanding the base of logarithms is important:

• The base of a logarithm determines how quickly the output of the function increases as the input increases.
• For the natural logarithm, a small increase in \(x\) results in a relatively larger increase in the logarithmic value compared to bases larger than \(e\).

This makes \(\ln(x)\) particularly valuable in calculus and theoretical mathematics, where it helps describe natural phenomena and provides solutions to differential equations.