Problem 85 - Precalculus with Calculus Previews

Question

In Problems \(85-88\), discuss how the graph of the given function can be obtained from the graph of \(f(x)\) \(=\ln x\) by means of a rigid transformation (a shift or a reflection)? $$ y=\ln 5 x $$

Step-by-Step Solution

Answer

The graph of \( y = \ln 5x \) is obtained by a horizontal compression of \( \ln x \) by a factor of 5.

Step 1: Understanding the Parent Function

The parent function given is \( f(x) = \ln x \). This is the natural logarithm function, which is defined only for \( x > 0 \) and has a vertical asymptote at \( x = 0 \). Its graph passes through the point (1, 0).

Step 2: Identifying the Transformation

The given transformation is \( y = \ln 5x \). This can be rewritten using the property of logarithms: \( y = \ln(5) + \ln(x) \). This indicates a horizontal compression of the graph of \( \ln x \).

Step 3: Applying the Horizontal Compression

The term \( \ln(5x) \) suggests that the graph of \( \ln x \) is horizontally compressed by a factor of \( \frac{1}{5} \) (i.e., points on the graph move closer to the y-axis by a factor of 5). This means that instead of the point \((1,0)\) on \( \ln x \), \( y = \ln 5x\) passes through the point \( \left( \frac{1}{5}, 0 \right) \).

Step 4: Final Graph Description

The graph of \( y = \ln 5x \) is obtained from \( f(x) = \ln x \) by compressing it horizontally by a factor of 5. This transformation does not involve any vertical shifts or reflections. The general shape of \( \ln x \) remains the same. It still has a vertical asymptote at \( x = 0 \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a fundamental mathematical function frequently used to solve problems involving exponential growth. It is defined for all positive \( x \) and has unique properties that make it an essential tool in algebra and calculus. The natural logarithm literally refers to the logarithm base of Euler's number (\( e \approx 2.71828 \)).
Understanding \( \ln x \) helps unravel exponential equations by converting multiplication into addition, which is easier to solve. The graph of \( f(x) = \ln x \) has some distinct characteristics:

• It passes through the point (1, 0), meaning \( \ln(1) = 0 \).
• Defined only for \( x > 0 \), making it non-existent for negative numbers and zero.
• The curve is always increasing, but the rate of increase slows down as \( x \) becomes larger.
• There is a vertical asymptote at \( x = 0 \), where the curve approaches but never actually touches the y-axis.

A firm grasp of the natural logarithm is crucial for handling transformations and understanding their effects on different functions.

Horizontal Compression

Horizontal compression involves shrinking a graph horizontally by a certain factor towards the y-axis. When you see a function like \( y = \ln 5x \), you're looking at a horizontal compression.
This transformation means every point on \( \ln x \) moves closer to the y-axis by a factor relative to the constant multiplied by \( x \), in this case, 5.

• To understand the effect, you rewrite \( \ln(5x) \) as \( \ln 5 + \ln x \) using logarithmic properties.
• This suggests that the usual point \( (1,0) \) on \( \ln x \) shifts to \( \left(\frac{1}{5}, 0\right) \) on \( \ln 5x \).
• The distance between points on the graph and the vertical asymptote decreases, compacting the graph horizontally.

Horizontal compression does not alter the general shape or direction of the curve; it just changes how spread out or concentrated the graph is along the x-axis, focusing points around the y-axis.

Vertical Asymptote

An important feature of logarithmic functions is the presence of a vertical asymptote. In the context of the natural logarithm, this vertical asymptote appears at \( x = 0 \).
Vertical asymptotes are lines that the graph approaches but never actually crosses or meets.

• For \( f(x) = \ln x \), as \( x \) approaches zero from the positive side, the value of \( \ln x \) drops steeply towards negative infinity.
• This behavior of shooting downward occurs because the logarithm of values close to zero results in increasingly negative outcomes.
• In transformations like \( \ln(5x) \), the vertical asymptote remains at \( x = 0 \). The horizontal compression does not shift the asymptote; it only modifies the curve's reach towards it.
• The presence of a vertical asymptote is pivotal for understanding the boundary behavior and possible solutions of logarithmic functions.

Grasping the concept of a vertical asymptote helps predict function behavior near boundaries and is crucial in sketching graphs accurately.