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Chapter 3: Problem 75

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

Short Answer

Expert verified
The domain of the function is \(x>4\), there is a vertical asymptote at \(x=4\), the x-intercept is at \(x=5\).

Step by step solution

01

Determine the Domain

The domain of \(f(x)\) is determined by the 'inside' of the function. The formula inside the natural logarithm function \(\ln(x-4)\) must be greater than zero, so \(x-4>0\). Solving this inequality will yield \(x>4\), which is the domain.
02

Find the x-intercepts

The x-intercepts of a function are the values of \(x\) where \(f(x)=0\). Set \(f(x)=0\) and solve for \(x\): \(\ln(x-4)=0\). Exponentiating both sides with a base of \(e\), we get \(x-4=1\), therefore, \(x=5\). So the x-intercept is \(x=5\).
03

Determine the vertical asymptote

For a logarithmic function, the vertical asymptote is at the 'inside' of the function when it equals zero. So in this case, we set \(x-4=0\), which gives us \(x=4\). So there is a vertical asymptote at \(x=4\).
04

Sketch the graph

Using the above information, a sketch of the graph can be created. The graph has a vertical asymptote at \(x=4\), an x-intercept at \(x=5\) and is defined for all values of \(x>4\).

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