Question

Make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. $$ y=\ln \left(\frac{1}{x}\right) $$

Short answer

Reflect the graph of \( y = \ln x \) across the x-axis.

Step-by-step solution

Understand the basic graph

The function \( y = \ln x \) is defined for \( x > 0 \). It is an increasing function that approaches \( -\infty \) as \( x \) approaches 0 from the right and increases without bound as \( x \) increases. The x-intercept occurs at \( x=1 \).

Analyze the transformation

The function given is \( y = \ln \left( \frac{1}{x} \right) \). To understand this, recall the logarithmic property \( \ln \left( \frac{1}{x} \right) = -\ln x \). This indicates that the graph of \( y = \ln \left( \frac{1}{x} \right) \) is a reflection of \( y = \ln x \) across the x-axis.

Determine key points

For \( x = 1 \), \( y = \ln \left( \frac{1}{1} \right) = \ln(1) = 0 \). This is the same as the natural log graph. For \( x > 1 \), \( \ln(x) > 0 \) so \( y = -\ln(x) < 0 \). Conversely, for \( 0 < x < 1 \), \( \ln(x) < 0 \), meaning \( -\ln(x) > 0 \). So, the graph will be above the x-axis for \( 0 < x < 1 \) and below it for \( x > 1 \).

Sketch the graph

Start by plotting the point at \( x = 1 \), \( y = 0 \). For \( x > 1 \), plot points where the function \( y \) decreases as \( x \) increases (the graph goes downward, reflecting over the x-axis from the original \( y = \ln x \) graph). For \( 0 < x < 1 \), the graph will increase towards the left as it approaches the y-axis.

Key concepts

Logarithmic Functions

Logarithmic functions are a crucial part of calculus, represented as the inverse of exponential functions. The function \( y = \ln x \) is the natural logarithm, where "ln" denotes logarithm to the base \( e \), with \( e \approx 2.71828 \). This function is only defined for positive \( x \) values because the logarithm of zero or a negative number is undefined.
Logarithms are used to solve equations where the variable is an exponent. Understanding their properties—such as \( \ln(ab) = \ln a + \ln b \) and \( \ln(a^b) = b \cdot \ln a \)—is vital when simplifying expressions. For instance, using the property \( \ln\left(\frac{1}{x}\right) = -\ln x \) can help us transform and analyze the function \( y = \ln\left(\frac{1}{x}\right) \). This inherent flexibility makes logarithmic functions a powerful tool in both pure and applied mathematics.

Reflecting Graphs

Reflecting graphs is a graphing technique used to change the position of a graph by flipping it over a specific axis. For the function \( y = \ln\left(\frac{1}{x}\right) \), using the property \( \ln\left(\frac{1}{x}\right) = -\ln x \), we realize that the graph is a reflection of the graph of \( y = \ln x \) across the x-axis.
In practical terms, reflection means altering all y-values in the original function. If the y-values are positive, they become negative and vice-versa. This action creates a mirror image of the graph along the chosen axis. Reflecting graphs over the x-axis is particularly useful when dealing with reciprocal functions, such as in this exercise. It helps visualize and predict changes in the graph's behavior while preserving the x-intercepts' positions.

Graphing Techniques

Mastering graphing techniques involves understanding transformations like translations, reflections, stretches, and compressions. In the context of logarithmic functions, knowing how these transformations affect the graph aids immensely in sketching functions like \( y = \ln \left( \frac{1}{x} \right) \).

• Translation: Shifts the graph horizontally or vertically without changing its shape.
• Reflection: Mirrors the graph across an axis, transforming y-values as needed.
• Stretch & Compression: Alter the graph's scale, making it steeper or flatter, by multiplying/dividing x or y values by a constant.

For the problem at hand, focusing on the reflection over the x-axis helps to understand how the behavior of \( y = \ln x \) is modified when graphed as \( y = \ln \left( \frac{1}{x} \right) \). The graph will now dive below the x-axis for \( x > 1 \) and rise above the x-axis for \( 0 < x < 1 \), informing both the sketch and the behavior of the function.

Natural Logarithm Properties

The natural logarithm, \( \ln x \), is a function that comes with several important properties. These properties provide the framework necessary for manipulating and understanding logarithmic functions. One of the core properties is the product rule: \( \ln(ab) = \ln a + \ln b \), which simplifies complex logarithmic expressions.
A particularly important property for this exercise is the reciprocal rule: \( \ln\left(\frac{1}{x}\right) = -\ln x \). This property is the key to understanding the transformation performed in the original exercise, where the function \( y = \ln x \) is reflected over the x-axis to become \( y = \ln\left(\frac{1}{x}\right) \).
Understanding these properties not only aids in graphing but also in solving equations, simplifying expressions, and conducting calculus operations involving logarithms. Utilizing these characteristics and transformations efficiently empowers students to handle a variety of mathematical challenges with confidence.