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Chapter 4: Problem 15

Sketch the graph of the function. $$ y=\ln 2 x $$

Short Answer

Expert verified
The graph of function \(y = \ln(2x)\) is a compressed version of the base \(\ln(x)\) function. It increases gently to the right, steeply to the left (for \(0 < x < 1\)), with a vertical asymptote at \(x=0\), and a range of all real numbers. The function is always increasing.

Step by step solution

01

Understand the Base Function

The graph of base function \(\ln x\) is such that at \(x=1\), \(y=0\) and it increases gently to the right, but steeply to the left between \(0 < x < 1\). There is a vertical asymptote at \(x=0\) and the range is all real numbers. The function is always increasing.
02

Understand the Transformation

The function to be graphed is \(\ln (2x)\), not \(\ln x\). The factor of 2 inside the function means that every 'x' in the base function is halved. This has the effect of horizontally compressing the graph of \(\ln x\) by a factor of 2.
03

Create the Graph

Plot the base function \(\ln x\) and compress it horizontally by a factor of 2 to produce the graph of \(\ln(2x)\). Note that as \(x\) approaches 0 from the right, \(y\) approaches \(-\infty\). This gives a vertical asymptote of \(x=0\). Also, as \(x\) becomes infinitely large, \(y\) gradually increases without bound.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Asymptote
A vertical asymptote is a key feature in many logarithmic functions, including \( y = \ln 2x \). It represents a line that the graph approaches but never touches or crosses. In the base function \( \ln x \), this asymptote is located at \( x = 0 \). This means the function is undefined for \( x \leq 0 \) and will approach infinity as it reaches this line from the right.

In our transformed function \( \ln(2x) \), the vertical asymptote remains at \( x = 0 \). Here's why:
  • The expression inside the logarithm \( 2x \) is set to zero for the asymptote. Solving \( 2x = 0 \) still gives \( x = 0 \).
  • The behavior near the asymptote means that as \( x \) approaches \( 0^+ \), \( y = \ln(2x) \) tends to \(-\infty\).
Understanding this helps in sketching the graph, as you know that the function will decrease sharply towards negative infinity as \( x \) nears zero from the right.
Horizontal Compression
Horizontal compression involves shrinking the graph towards the y-axis. For the function \( \ln(2x) \), this effect occurs due to the factor of 2 inside the logarithm. This means each x-value on the graph of \( \ln x \) is halved, pulling the graph closer to the y-axis.

Here's how it works:
  • In the transformation, every \( x \) value in the base graph is replaced by \( x/2 \) in \( \ln x \). Therefore, to have the same effect, \( \ln(2x) \) compresses the graph by a factor of 2.
  • As a result, key points like \( x = 1 \), which was originally \( y = 0 \), moves to \( x = 0.5 \) but remains \( y = 0 \).
This horizontal compression does not affect the vertical asymptote or the range of the function, which remains all real numbers. It does, however, make the graph steeper as you move away from the y-axis.
Base Function Transformation
Transforming base logarithmic functions such as \( \ln x \) is crucial to graphing \( \ln(2x) \). Transformations modify the graph through translations, stretches, and compressions.

For \( \ln(2x) \), the primary transformation is the horizontal compression. This transformation alters only the x-coordinates:
  • The base logarithm \( \ln x \) tells us how the function behaves. Key point: \( (1, 0) \) remains the same after transformation in terms of y-height but shifts horizontally due to compression.
  • Each point \( (x, y) \) in the original graph translates to \( (x/2, y) \) in the transformed graph.
Recognizing these transformations allows you to faithfully sketch the graph of a transformed logarithm efficiently.
Remember, while the compression affects the width of the graph, it does not change the asymptotic nature or overall range.

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