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Chapter 3: Problem 38

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
The domain of the function \(g(x)=\log _{6} x\) is (0, +鈭�). The x-intercept is 1. The vertical asymptote is \(x=0\).

Step by step solution

01

Find the Domain

The domain of a function is the set of all possible input values (x-values) which will output real numbers. For the function \(g(x)=\log _{6} x\), since you cannot take the log of a negative number or zero, the domain of \(g\) is \((0,+\infty)\). That means \(x\) can only be positive values only and also \(x\) cannot be 0.
02

Determine the X Intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. So it's where \(g(x) = 0\). We solve the equation \(0=\log _{6} x\), which tells us when our \(y\) equals 0, what is our \(x\). Solving the equation for \(x\) will result in \(x=1\). So the x-intercept is \(1\).
03

Finding the Vertical Asymptote

Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. They can be considered 'boundaries' that the graph approaches but does not cross. In this case, \(x=0\) is the vertical asymptote as the function approaches infinity on the y-axis the closer \(x\) gets to 0 but never touches or crosses \(x=0\).
04

Sketch the Graph

Plot the asymptote at \(x=0\). Plot the x-intercept at \(x=1\). Then sketch the curve that approaches the asymptote as \(x\) gets closer to 0, and passes through the intercept.

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