/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Graph each function. $$f(x)=\l... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 40

Graph each function. $$f(x)=\log _{10} x$$

Short Answer

Expert verified
Graph is a curve with vertical asymptote at \(x=0\), passing through \((1,0)\), \((10,1)\), and \((0.1,-1)\).

Step by step solution

01

Understanding the Logarithmic Function

The function given is a logarithmic function with base 10, written as \(f(x) = \log_{10} x\). This means that for a given \(x\), \(f(x)\) will return the exponent to which the base 10 must be raised to yield \(x\).
02

Identifying Domain and Range

For the function \(f(x) = \log_{10} x\), the domain is all positive real numbers, \(x > 0\), since the logarithm is undefined for zero and negative numbers. The range is all real numbers, \(-\infty < f(x) < \infty\).
03

Finding Key Points

To graph the function, we need some key points. Since \(\log_{10} 1 = 0\), we have the point \((1, 0)\). For \(\log_{10} 10 = 1\), we get \((10, 1)\). For \(\log_{10} 100 = 2\), we get \((100, 2)\). Similarly, for \(\log_{10} 0.1 = -1\), we get \((0.1, -1)\).
04

Drawing the Graph

On a coordinate plane, plot the points identified: \((1, 0)\), \((10, 1)\), \((100, 2)\), \((0.1, -1)\). The graph will approach the y-axis but never touch it, creating a vertical asymptote at \(x = 0\). The graph will continue to rise through the plotted points.
05

Sketching the Shape of the Graph

The logarithmic curve starts from the right side of the y-axis, moving upwards to the right. As \(x\) approaches zero from the right, \(f(x)\) decreases towards negative infinity. As \(x\) increases, \(f(x)\) increases but at a decreasing rate, illustrating the slow growth of logarithmic functions.

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Key Concepts

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

Domain and Range
The domain of a logarithmic function is an essential concept to grasp. For the function \(f(x) = \log_{10} x\), the domain consists of all positive real numbers. This is because logarithms are undefined for zero and negative values. Thus, when graphing \(f(x)\), we only consider points where \(x > 0\).

In terms of range, the function \(f(x) = \log_{10} x\) can produce any real number. This means that as \(x\) increases or decreases within its domain, \(f(x)\) can take on any value from \(-\infty\) to \(\infty\).
This complete set of real numbers as the range is a distinctive feature of logarithmic functions, allowing their graphs to stretch infinitely in the vertical direction.
Graphing Techniques
Graphing a logarithmic function like \(f(x) = \log_{10} x\) involves identifying key characteristics and points. Start by picking a few values of \(x\) that result in simple calculations of \(f(x)\).

For instance:
  • When \(x = 1\), \(f(x) = \log_{10} 1 = 0\)
  • When \(x = 10\), \(f(x) = \log_{10} 10 = 1\)
  • When \(x = 100\), \(f(x) = \log_{10} 100 = 2\)
  • When \(x = 0.1\), \(f(x) = \log_{10} 0.1 = -1\)
Once these points are plotted on a coordinate plane, you can start connecting them to see the logarithmic curve form. The line will rise smoothly from left to right, emphasizing the gentle increase associated with logarithmic growth. Making several precise plots can help ensure an accurate representation of the function鈥檚 behavior.
Asymptotes
An important characteristic of the graph of \(f(x) = \log_{10} x\) is the presence of a vertical asymptote. This asymptote is located at \(x = 0\), which means the graph approaches the y-axis, but never intersects or touches it.

This behavior results from the undefined nature of logarithms at zero and negative values. Consequently, as \(x\) approaches zero from the right, the function \(f(x)\) tends toward negative infinity.
The graphical implication of this is that the curve will get closer and closer to the vertical line \(x = 0\) but will never meet it. Understanding this aspect helps in accurately sketching the function and is a hallmark of all logarithmic graphs.

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Most popular questions from this chapter

The revenue in millions of dollars for the first 5 years of Internet advertising is given by \(A(x)=25(2.95)^{x},\) where \(x\) is years after the industry started. (Source: Business Insider.) (a) What was the Internet advertising revenue after 5 years? (b) Determine analytically when revenue was about \(\$ 250\) million. (c) According to this model, when did the Internet advertising revenue reach \(\$ 1\) billion?

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. \(\log _{10} x=x-2\)

Restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. $$f(x)=x^{4}$$

In 1666 the village of Eyam, located in England, experienced an outbreak of the Great Plague. Out of 261 people in the community, only 83 survived. The table shows a function \(f\) that computes the number of people who had not (yet) been infected after \(x\) days. $$\begin{array}{|c|r|r|r|r|}\hline x & 0 & 15 & 30 & 45 \\\\\hline f(x) & 254 & 240 & 204 & 150 \\\\\hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline x & 60 & 75 & 90 & 125 \\\\\hline f(x) & 125 & 103 & 97 & 83\\\\\hline\end{array}$$ (a) Use a table to represent a function \(g\) that computes the number of people in Eyam who were infected after \(x\) days. (b) Write an equation that shows the relationship between \(f(x)\) and \(g(x)\) (c) Use graphing to decide which equation represents \(g(x)\) better \(y_{1}=\frac{171}{1+18.6 e^{-0.0747 x}}\) or \(y_{2}=18.3(1.024)^{x}\) (d) Use your results from parts (b) and (c) to find a formula for \(f(x)\)

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1. Assume that all variables represent positive real numbers. $$\ln (a+b)+\ln a-\frac{1}{2} \ln 4$$
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