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Chapter 3: Problem 66

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 \(h(x)=\ln(x+5)\) is \((-5, \infty)\), the x-intercept is at \(-4\), and there is a vertical asymptote at \(x=-5\).

Step by step solution

01

Finding the Domain of the Logarithmic Function

Start by finding the domain of the logarithmic function \(h(x)= \ln (x+5)\). The domain is the set of all possible x-values. The logarithm of a negative number or zero is undefined, so we set \(x+5>0\). Solving this inequality gives \(x > -5\). So, the domain of the function is \(-5 < x < \infty\) or in interval notation \((-5, \infty)\)
02

Finding the x-intercept of the function

The x-intercept is the value of x at which the function equals zero. Thus, we set \(h(x)=0\) and solve for \(x\). Therefore, we get \(0=\ln(x+5)\). This can be changed to exponential form as \(e^0=x+5\). Hence, we get the x-intercept as \(x=e^0-5=1-5=-4\). Thus, the x-intercept is at \(-4\).
03

Finding the Vertical Asymptote

The logarithmic function \(h(x)=\ln(x+5)\) will have a vertical asymptote at \(x=-5\) because as we approach this x-value from the right, the function increases without bound.
04

Graphing the Function

On the graph, mark the x-intercept at \(-4\), the vertical asymptote at \(x=-5\). The function starts from the vertical asymptote and passes through the x-intercept. The line rises to the right of the y-axis and approaches the x-axis but never crosses it. This provides a sketch of the function \(h(x)=\ln(x+5)\).

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Most popular questions from this chapter

Graphical Analysis Use a graphing utility to graph \(f\) and \(g\) in the same viewing window and determine which is increasing at the greater rate as \(x\) approaches + \(\infty\). What can you conclude about the rate of growth of the natural logarithmic function? (a) \(f(x)=\ln x, \quad g(x)=\sqrt{x}\) (b) \(f(x)=\ln x, \quad g(x)=\sqrt[4]{x}\)

Limit of a Function (a) Complete the table for the function \(f(x)=(\ln x) / x\)\begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 5 & 10 & 10^{2} & 10^{4} & 10^{6} \\ \hline f(x) & & & & & & \\ \hline \end{array}

A classmate claims that the following are true. (a) \(\ln (u+v)=\ln u+\ln v=\ln (u v)\) (b) \(\ln (u-v)=\ln u-\ln v=\ln \frac{u}{v}\) (c) \((\ln u)^{n}=n(\ln u)=\ln u^{n}\) Discuss how you would demonstrate that these claims are not true.

The graph of \(f(x)=\log _{3} x\) contains the point \((27,3)\)

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. $$x=e^{-5 / 6}$$
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