Question

Make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. $$ y=\ln \sqrt{x} $$

Short answer

The graph of \( y = \ln \sqrt{x} \) is a vertical compression of \( y = \ln x \) by a factor of 2.

Step-by-step solution

Recognize the Known Graph

The graph of the function \( y = \ln x \) is a curve that starts from the point (1,0) and increases slowly as \( x \) increases. It is defined for \( x > 0 \) and approaches negative infinity as \( x \) approaches 0.

Rewrite the New Function

The given function is \( y = \ln \sqrt{x} \). Remember that \( \sqrt{x} \) can be rewritten as \( x^{1/2} \). Using logarithmic properties, \( \ln \sqrt{x} = \ln (x^{1/2}) = \frac{1}{2} \ln x \).

Transform the Known Graph

Since the given function \( y = \ln \sqrt{x} \) simplifies to \( y = \frac{1}{2} \ln x \), this indicates that the graph of \( y = \ln x \) is vertically compressed by a factor of 2. This new graph will still start at the point (1,0) but will rise at half the speed.

Sketch the Graph

Start with the original graph \( y = \ln x \). For each point on this graph, such as (2, \ln 2), plot the point (2, \frac{1}{2} \ln 2) for the new graph \( y = \ln \sqrt{x} \). The overall shape remains the same, but every point on the original graph is now half as high on the y-axis.

Key concepts

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting a given graph. In this exercise, we specifically look at a **vertical compression**, which is a transformation where the graph is squished towards the x-axis.

When we have a function like \( y = \ln x \) and apply a coefficient to it, such as in the function \( y = \frac{1}{2} \ln x \), this scales the function vertically. It takes each point on the graph and moves it closer to the x-axis if the coefficient is between 0 and 1 (in this case, 1/2).

A vertical compression does not affect the x-values of the points on the graph, only the y-values. So if you started with a point \((a, \ln a)\) on the original graph, after the transformation, this point will become \((a, \frac{1}{2}\ln a)\). The transformation is visually evident since the graph remains the same shape but rises more slowly, indicating that it’s compressed vertically.

Properties of Logarithms

Logarithms are mathematical siblings of exponents and they have distinctive properties that allow us to manipulate expressions easily. One important property used in this exercise is \( \ln(a^b) = b \cdot \ln(a) \). Using this, we simplified the expression \( \ln \sqrt{x} \) into \( \ln(x^{1/2}) = \frac{1}{2} \ln x \).

This property shows how logarithms and powers interact, revealing that the exponent can be moved as a coefficient in front of the logarithm. It's useful in rewriting logarithmic expressions that involve roots, as roots can be expressed as fractional exponents.

Other important logarithm properties include:

• \( \ln(1) = 0 \)
• \( \ln(ab) = \ln a + \ln b \)
• \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)

These properties provide a powerful toolkit for simplifying and manipulating logarithmic expressions, making them integral to solving various mathematical problems.

Vertical Compression

Vertical compression is a type of transformation that changes the steepness of the graph of a function. With vertical compression, the function's graph is scaled down in the y-direction, which means the graph is pressed closer to the x-axis.

In the context of the given problem, we see a vertical compression where the function \( y = \ln x \) is modified by a factor of 1/2, resulting in \( y = \frac{1}{2}\ln x \). This implies that each y-coordinate on the graph of \( \ln x \) is halved, effectively compressing the graph vertically.

To visualize this concept, imagine taking a flexible graph and pressing it downward evenly at every point along the y-axis. Key characteristics of vertical compression include:

• The x-intercept remains the same.
• The overall shape of the graph does not change.
• The graph extends horizontally at a faster rate compared to its vertical rise due to the compressive effect.

Understanding vertical compression helps in sketching graphs accurately and in anticipating how a function's graph will look after such transformations.