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Chapter 3: Problem 37

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$f(x)=\log _{4} x$$

Short Answer

Expert verified
The domain of the function \(f(x)=\log _{4} x\) is \(x > 0\) or \((0, \infty)\). The x-intercept is at \(x=1\), and there is a vertical asymptote at \(x=0\).

Step by step solution

01

Determine the Domain

The logarithm \(\log_{b} x\) is defined only for \(x > 0\). Therefore, the domain of the function is \(x > 0\), or in interval notation, that is \((0, \infty)\).
02

Find the x-intercept

The x-intercept of the function is the value of \(x\) for which the function evaluates to 0. \nThis means we must solve the equation \(\log_{4} x = 0\) for \(x\). The x-intercept is found by realizing that any logarithm base \(b\) of 1 is 0, because any number raised to 0 is equal to 1. So, \(x=1\) is the x-intercept of the function.
03

Identify the Vertical Asymptote

Because the logarithmic function is not defined for \(x \leq 0\), there is a vertical asymptote at \(x=0\).
04

Sketch the Graph

As evident from steps 1 through 3, the graph of the function \(\log_{4} x\) goes through the point \((1,0)\), and it gets arbitrarily close to the line \(x=0\) (the y-axis). The graph grows slowly as \(x\) goes to infinity, and it descends indefinitely as \(x\) approaches 0 from the right.

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