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Chapter 10: Problem 16

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

Short Answer

Expert verified
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 \(x 鈮� 0\), and approaches \(y = 5\) as \(x\) approaches 0 from the right.

Step by step solution

01

Understand the basic logarithmic function

The basic natural logarithm function, \(y = \ln x\), is defined for \(x > 0\) and undetermined for \(x 鈮� 0\). 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.
02

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.
03

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 \(x 鈮� 0\).

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Key Concepts

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

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.

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Most popular questions from this chapter

In Exercises, find the derivative of the function. $$ y=x^{2} e^{x}-2 x e^{x}+2 e^{x} $$

The cost of producing \(x\) units of a product is modeled by \(C=100+25 x-120 \ln x, \quad x \geq 1\) (a) Find the average cost function \(\bar{C}\). (b) Analytically find the minimum average cost. Use a graphing utility to confirm your result.

In Exercises, use a graphing utility to verify that the functions are equivalent for \(x>0\). $$ \begin{aligned} &f(x)=\ln \frac{x^{2}}{4} \\ &g(x)=2 \ln x-\ln 4 \end{aligned} $$

On the Richter scale, the magnitude \(R\) of an earthquake of intensity \(I\) is given by \(R=\frac{\ln I-\ln I_{0}}{\ln 10}\) where \(I_{0}\) is the minimum intensity used for comparison. Assume \(I_{0}=1\). (a) Find the intensity of the 1906 San Francisco earthquake for which \(R=8.3\). (b) Find the intensity of the May 26, 2006 earthquake in Java, Indonesia for which \(R=6.3\). (c) Find the factor by which the intensity is increased when the value of \(R\) is doubled. (d) Find \(d R / d I\)

The term \(t\) (in years) of a \(\$ 200,000\) home mortgage at \(7.5 \%\) interest can be approximated by \(t=-13.375 \ln \frac{x-1250}{x}, x>1250\) where \(x\) is the monthly payment in dollars. (a) Use a graphing utility to graph the model. (b) Use the model to approximate the term of a home mortgage for which the monthly payment is \(\$ 1398.43 .\) What is the total amount paid? (c) Use the model to approximate the term of a home mortgage for which the monthly payment is \(\$ 1611.19 .\) What is the total amount paid? (d) Find the instantaneous rate of change of \(t\) with respect to \(x\) when \(x=\$ 1398.43\) and \(x=\$ 1611.19\). (e) Write a short paragraph describing the benefit of the higher monthly payment.
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