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Chapter 3: Problem 42

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $g(x)=\log _{6} x$$

Short Answer

Expert verified
Domain of \(g(x) = \log_{6}{x}\) is \(x > 0\). X-intercept is \(x = 1\). Vertical asymptote is \(x = 0\). The graph starts from the bottom left near the y-axis and rises to cross the x-axis at \(x = 1\), then continues to rise to the right at a decreasing rate. It gets very close but never touches the y-axis due to its vertical asymptote.

Step by step solution

01

Finding the domain

The domain of a function is the set of all possible x-values, or inputs, for which the function is defined. The domain of the logarithmic function \(g(x) = \log_{6}{x}\) is \(x > 0\). This is because we can only take logs of positive numbers.
02

Finding the x-intercept

The x-intercept is the x-coordinate at which the function intersects the x-axis. This happens when the output, or y-value, of the function is zero. Setting \(g(x) = 0\), we have \(\log_{6}{x} = 0\). Solving for \(x\), we use the property that \(a^{0} = 1\) for any nonzero number \(a\), in this case, \(a = 6\). Therefore, \(x = 1\).
03

Finding the vertical asymptote

A vertical asymptote of a function is a vertical line \(x = a\) where the function approaches infinity or negative infinity as \(x\) approaches \(a\) from the left or right. For the logarithmic function \(g(x) = \log_{6}{x}\), as \(x\) approaches zero from the right (meaning, as \(x\) gets smaller and smaller but remains positive), \(\log_{6}{x}\) decreases without bound. Therefore, the vertical asymptote of the function is \(x = 0\).
04

Sketching the graph

From the previous steps, we know that the function is defined for \(x > 0\), crosses the x-axis at \(x = 1\), and has a vertical asymptote at \(x = 0\). Given these, the function starts from the bottom left-hand side of the graph (near the y-axis) and rises to cross the x-axis at \(x = 1\), continues to rise to the right but at a decreasing rate. Due to its negative infinity value as \(x\) approaches from the right of \(x=0\), it gets very close but never touches the y-axis.

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Most popular questions from this chapter

In Exercises \(103-106,\) use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 4} x$$

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