Problem 17 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, sketch the graph of the function. $$ y=3 \ln x $$

Step-by-Step Solution

Answer

The graph passes through (1,0), vertically stretched compared to the simple \( \ln x \) function due to the multiplication by 3, and has a vertical asymptote at the y-axis.

Step 1: Plot the Basic Logarithmic Function

Start by sketching the graph of the basic logarithm function \( y=\ln x \). This function has a vertical asymptote at x=0 (the y-axis), it passes through the point (1,0), and continues to increase as x gets larger.

Step 2: Apply the Vertical Stretch

The '3' in the equation \( y=3 \ln x \) means every y-value of the basic logarithmic function is multiplied by 3. This causes the graph to stretch vertically. To represent this, multiply the y-values of some specific points on the basic log graph by 3, such as (1, 0), (2,0.69), (3,1.1) and re-plot these new points

Step 3: Sketch the Final Graph

Draw a smooth curve that passes through the points plotted in Step 2 and approaches but never crosses the y-axis. This is the graph of \( y = 3 \ln x \)

Key Concepts

Graph Sketching

Graph sketching is an essential skill in understanding how functions behave visually. When sketching the graph of a function like the natural logarithm function, you start with its basic properties. For the function \( y = \ln x \), the graph has a vertical asymptote at \( x = 0 \). This means the curve will never actually touch or cross the y-axis. Remember, the y-values (outputs) increase as the x-values (inputs) get larger, reflecting the nature of logarithmic growth.
To trace this graph, mark a notable point: \( (1, 0) \). Here, the function equals zero because \( \ln 1 = 0 \). The graph will pass through this point and continue to rise slowly to the right.

• The graph approaches negative infinity as x approaches zero from the right.
• The graph increases but flattens out as x becomes larger.

When starting your sketch, it's good practice to plot a few additional points like \( (2, 0.69) \) or \( (3, 1.1) \). This gives you a more accurate idea of the curve's shape. Drawing a smooth curve through these points helps you visualize the function's behavior.

Vertical Stretch

A vertical stretch changes the shape of a graph by increasing the y-coordinates of all points by a specific factor. This adjustment allows you to visually communicate the impact of multiplying a function by a constant. In the equation \( y = 3 \ln x \), the log function is stretched vertically by a factor of 3. This means every point on the basic \( \ln x \) graph moves farther away from the x-axis by three times its original distance.

• For example, the point \( (2, 0.69) \) becomes \( (2, 3\times 0.69) = (2, 2.07) \).
• The point \( (3, 1.1) \) transforms to \( (3, 3\times 1.1) = (3, 3.3) \).

This scaling widens the gap between the graph's curve and the x-axis without altering the x-values themselves. In mathematical terms, vertical stretching multiplies each output value by a given factor, extending the graph vertically. Remember, the vertical stretch does not affect the x-values, only the y-values are scaled.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm whose base is the mathematical constant \( e \approx 2.718 \). It serves a crucial role in mathematics due to its properties and widespread applications.

• The function \( y = \ln x \) tells us how many times we would multiply \( e \) to get x.
• It is commonly used in calculus for integration and differentiation because of its simple derivative, \( \frac{d}{dx}(\ln x) = \frac{1}{x} \).

The range of \( \ln x \) is all real numbers, while its domain is the positive real numbers \( x > 0 \). The natural logarithm grows slower than polynomial or exponential functions, yet is faster than linear functions for large x-values. Understanding \( \ln x \) is pivotal when working with exponential growth models, compound interest in finance, or continuous compounding in various fields. Its properties make it a powerful tool in both theoretical and applied mathematics. By grasping \( \ln x \)'s role, you enhance your mathematical toolbox for solving complex problems.