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Chapter 3: Problem 76

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
Domain: \(x > -5\), x-intercept: x = -4, Vertical Asymptote: \(x = -5\)

Step by step solution

01

Finding the Domain

The domain of a logarithmic function is the set of x-values for which the function is defined. For our function \(h(x) = \ln(x+5)\), the argument of the logarithm (i.e. the expression inside the logarithm) must be positive. This means that we must have \(x+5 > 0\). Solving this inequality gives us \(x > -5\). So, the domain of the function is \(x > -5\).
02

Finding the x-Intercept

The x-intercept of a function is the point where the graph of the function crosses or touches the x-axis. This point occurs when the value of the function is zero. Therefore, to find the x-intercept, we set our function equal to zero and solve for x. So, we have \(0 = \ln(x+5)\). Using the property of logarithms that says the equation \(\ln(x) = 0\) if and only if \(x = 1\), we find that \(x + 5 = 1\), which implies \(x = -4\). So, the x-intercept of the function is at x = -4.
03

Finding the Vertical Asymptote

The vertical asymptote of a logarithmic function is the line \(x = a\), where \(a\) is the x-coordinate where the graph touches or crosses the x-axis. For our function, the graph touches the x-axis at \(x = -4\), but the graph is undefined at \(x = -5\), since that would make the argument of the logarithm zero (in logarithm, the argument should be greater than 0). Therefore, the vertical asymptote is at \(x = -5\) which is a boundary point of the domain.
04

Sketching the Graph

To sketch the graph of \(h(x) = \ln(x+5)\), we plot the x-intercept at \(x = -4\) and the vertical asymptote at \(x = -5\). Since the function is logarithmic, the portion of the graph to the right of the asymptote rises gradually from left to right. The portion of the graph to the left of the asymptote approaches the asymptote but never quite touches it.

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{2} x+\log _{2}(x+2)=\log _{2}(x+6)$$

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