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Chapter 3: Problem 39

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$y=-\log _{3} x+2$$

Short Answer

Expert verified
The domain of the function is \(x>0\), it has an x-intercept at \(x=9\) and a vertical asymptote at \(x=0\).

Step by step solution

01

Finding the Domain

The domain of a logarithm function \(y = \log_{b} {x} \) is all real numbers \(x > 0 \). So the domain of the given function will be all real numbers \(x > 0\).
02

Finding the X-intercepts

The x-intercept occurs when \(y = 0\) in the equation of the function. So, we set \(y = 0\) in the given function \(0=-\log _{3} x+2\). This simplifies to \(\log _{3} x = 2\). We know that \(\log_b a = c\) is equivalent to \(b^c = a\). So \(3^2 = x\), giving us \(x = 9\). So the x-intercept of the function is at x=9.
03

Find the Vertical Asymptote

For the logarithmic function \(y = \log_b x\), the vertical asymptote is always at \(x=0\). Hence, for the given function as well, the vertical asymptote is at \(x=0\).
04

Sketch the graph

First plot the x-intercept at \(x=9\), and then draw a vertical line at \(x=0\) to represent the vertical asymptote. Since the log function is decreasing (due to the negative sign), the graph will be decreasing and approaching the vertical asymptote, but never crossing it. Similarly, from the x-intercept, the graph will also decrease and go to negative infinity as \(x\) increases.

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