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Chapter 12: Problem 104

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\log x, g(x)=-\log x$$

Short Answer

Expert verified
Function \(g(x)=-\log{x}\) is a vertical reflection of function \(f(x)=\log{x}\) over the x-axis.

Step by step solution

01

Graph the Function \(f(x) = \log{x}\)

To begin, graph the first function, \(f(x) = \log{x}\), in the viewing rectangle. This function represents a basic logarithm, so it should be fairly straightforward to plot. The function will start near negative infinity when \(x\) approaches 0 from the right and will pass through the point (1,0), growing slowly as \(x\) increases.
02

Graph the Function \(g(x) = -\log{x}\)

In the same viewing rectangle, graph the second function \(g(x) = -\log{x}\). The negative sign means that this function will be a reflection of \(f(x)\) over the x-axis. It will start near positive infinity when \(x\) approaches 0 from the right, will pass through the point (1,0), and will decrease as \(x\) increases.
03

Describe the Relationship

With both functions plotted, the relationship becomes clear. Function \(g(x)\) is a vertical reflection of function \(f(x)\) over the x-axis. This is clearly visible in the graph, as \(f(x)=\log{x}\) is flipped upside down to match with \(g(x)=-\log{x}\).

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Most popular questions from this chapter

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\ln x, g(x)=\ln (x+3)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\ln \sqrt{2}=\frac{\ln 2}{2}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When graphing a logarithmic function, I like to show the graph of its horizontal asymptote.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I expanded \(\log _{4} \sqrt{\frac{x}{y}}\) by writing the radical using a rational exponent and then applying the quotient rule, obtaining \(\frac{1}{2} \log _{4} x-\log _{4} y\)
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