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Chapter 3: Problem 83

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window. $$f(x)=\ln x+8$$

Short Answer

Expert verified
The graph of the function \(f(x) = \ln(x) + 8\) is a shifted version of the standard natural logarithm function, \(\ln(x)\), moved 8 units upward. It starts at the point (1,8) and increases slowly without crossing the x-axis. An appropriate viewing window for this function on a graphing utility could be x: [0.1, 10] and y: [-2, 10].

Step by step solution

01

Understanding the Function

The given function is \(f(x) = \ln(x) + 8\). This function is the natural logarithm function shifted vertically by 8 units. It's important to note that the natural log function, \(\ln(x)\), is undefined for \(x \leq 0\), therefore the domain of \(f(x)\) is \((0, \infty)\). This implies that the viewing window should start from slightly above 0 in the x-direction.
02

Graphing the Function

Use a graphing utility to graph the function. An appropriate viewing window would be from slightly above 0 to some positive number in the x-direction (for example, [0.1, 10]) and, in the y-direction, enough to display the value of 8 (for example, [-2, 10]). This range should display the curve of the logarithmic function entirely, including the vertical shift by 8 units.
03

Analyzing the Graph

Once the function is graphed, it should be a curve rising from the point (1,8), which is the shift from the usual start point of (1,0) for the basic \(\ln(x)\) function. The graph should increase slowly and should not cross the x-axis, as it is shifted 8 units up.

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Most popular questions from this chapter

(a) solve for \(P\) and (b) solve for \(t\). $$A=P e^{r t}$$

The values \(y\) (in billions of dollars) of U.S. currency in circulation in the years \(\begin{array}{lllll}2000 & \text { through } 2007 & \text { can be } & \text { modeled } & \text { by }\end{array}\) \(y=-451+444 \ln t, 10 \leq t \leq 17,\) where \(t\) represents the year, with \(t=10\) corresponding to 2000 . During which year did the value of U.S. currency in circulation exceed \$690 billion? (Source: Board of Governors of the Federal Reserve System)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{3} x+\log _{3}(x-8)=2$$

Let \(f(x)=\log _{a} x\) and \(g(x)=a^{x},\) where \(a>1\) (a) Let \(a=1.2\) and use a graphing utility to graph the two functions in the same viewing window. What do you observe? Approximate any points of intersection of the two graphs. (b) Determine the value(s) of \(a\) for which the two graphs have one point of intersection. (c) Determine the value(s) of \(a\) for which the two graphs have two points of intersection.

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1-\ln x}{x^{2}}=0$$
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