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Chapter 3: Problem 67

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$h(x)=\ln (x+5)$$

Short Answer

Expert verified
The domain of the function is \(x\in(-5,\infty)\), the x-intercept is at \(x=-4\), and the vertical asymptote is \(x=-5\). The graph is a curve originating from the vertical line \(x=-5\) (the vertical asymptote), crossing the x-axis at \(x=-4\), and going upwards to the right.

Step by step solution

01

Find the Domain

The domain of a logarithmic function is the set of positive real numbers. In this case, we look for values of \(x\) for which \(x+5>0\). Solving for \(x\) gives \(x>-5\). So, the domain of the function is \(x\in(-5,\infty)\).
02

Find the \(x\)-intercept

The \(x\)-intercept of a function is the point where the function crosses or touches the x-axis. For this, we set \(h(x)\) equal to zero and solve for \(x\). So, we have: \[0=\ln(x+5)\]. Converting this logarithmic equation to an exponential equation gives that \(e^{0}=x+5\), which simplifies to \(1=x+5\). From that, we find that \(x=-4\). So, the \(x\)-intercept is \(x=-4\).
03

Find the Vertical Asymptote

A vertical asymptote of a function is a vertical line \(x=a\) such that \(f(x)\) approaches infinity as \(x\) approaches \(a\). For a logarithmic function of the form \(\ln(x+a)\), the vertical asymptote is found where \(x + a = 0\). Therefore, the vertical asymptote of the function \(h(x)=\ln (x+5)\) is \(x = -5\).
04

Sketch the Graph of Function

To sketch the graph of the function \(h(x)=\ln (x+5)\), note the following: The function is shifted 5 units to the left compared to the standard natural logarithm function. The graph will not exist for \(x < -5\). And at \(x = -4\), the graph crosses the x-axis. Draw a curve originating from the vertical asymptote (\(x = -5\)) and passing through the point (-4,0). The curve increases as \(x\) moves to the right towards infinity.

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