Problem 16 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, sketch the graph of the function. $$ y=5+\ln x $$

Step-by-Step Solution

Answer

The graph of the function \(y = 5 + \ln x\) is similar to that of \(y = \ln x\) but shifted five units upward. It crosses the x-axis at x=1 at the point of (1,5), is undefined for \(x ≤ 0\), and approaches \(y = 5\) as \(x\) approaches 0 from the right.

Step 1: Understand the basic logarithmic function

The basic natural logarithm function, \(y = \ln x\), is defined for \(x > 0\) and undetermined for \(x ≤ 0\). Its graph passes through the point (1,0), increases without bound as \(x\) approaches infinity, and decreases without bound as \(x\) approaches 0 from the right.

Step 2: Understand the effect of transformations

Adding a constant to a function results in a vertical shift of its graph. Here, adding 5 to \( \ln x\) shifts the graph of the function five units upward.

Step 3: Sketch the transformed graph

To sketch the graph of \(y = 5 + \ln x\), start with the basic graph of \(y = \ln x\). Then, shift every point on that graph five units up. The graph still passes through the vertical line \(x=1\), but now at the point of (1,5), and is undefined for \(x ≤ 0\).

Key Concepts

Logarithmic Transformations

Logarithmic transformations play a crucial role in modifying the shape and position of logarithmic graphs. When we talk about transformations, there are a few kinds that affect graphs, such as reflections, stretches, shrinks, and shifts.
For the logarithmic function, the most common transformations include:

• Horizontal Shifts: Moving the graph left or right along the x-axis. This is usually achieved by adding or subtracting a constant inside the logarithm, such as \( \ln(x-c) \).
• Vertical Stretches/Shrinks: Multiplying the whole function by a constant to stretch or compress it vertically.

Understanding these fundamental transformations helps in visualizing the graph's movement without drawing it absolutely from scratch every time a change is made.

Vertical Shifts

A vertical shift involves moving the graph of a function up or down along the y-axis. This can be done by adding or subtracting a constant to/from the function itself.
In our specific example, adding "5" to \( \ln(x) \) results in the equation \( y = 5 + \ln x \). This means:

• Graph Movement: The entire graph of \( \ln x \) is shifted vertically upward by 5 units.
• Graph Characteristics: Its behavior and end-points remain the same; it just shifts position.

Such vertical shifts do not alter the "shape" of the graph, only its location on the coordinate plane.

Natural Logarithm Properties

Natural logarithms, denoted as \( \ln x \), have some unique properties due to their base \( e \), which is an irrational constant approximately equal to 2.718.
Some key properties include:

• Domain: \( x>0 \), meaning values of x must be positive since logarithms of non-positive numbers aren't defined in the context of real numbers.
• Range: All real numbers \((-\infty, \infty)\), as the output of a natural log can be any real value.
• Key Point: The natural logarithm function passes through the point (1,0) because \( \ln(1) = 0 \).
• Behaviour Near Zero: As \( x \) approaches 0 from the right, \( \ln x \) decreases without bound, heading towards minus infinity.

Understanding these properties aids in recognizing how the function behaves and what transformations like the ones above imply for its overall graph.