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Chapter 6: Problem 1

What are the domain and range of \(\ln x ?\)

Short Answer

Expert verified
Answer: The domain of \(\ln x\) is (0, \infty) and the range is (-\infty, \infty).

Step by step solution

01

Understanding the natural logarithm function

The natural logarithm function, denoted as \(\ln x\), is the inverse of the exponential function with base \(e\), \(e^x\). In other words, if \(y = \ln x\), then \(x = e^y\).
02

Find the domain of \(\ln x\)

The domain of a function is the set of all possible inputs (x-values) for which the function is defined. Since the natural logarithm function is the inverse of the exponential function, it is not defined for negative values of \(x\) or when \(x=0\) because \(e^x\) can never be equal to a negative number or 0. Therefore, the domain of \(\ln x\) is all positive real numbers: Domain: \((0, \infty)\)
03

Find the range of \(\ln x\)

The range of a function is the set of all possible output values (y-values) for which the function is defined. Given that the function \(y = \ln x\) is the inverse of \(x = e^y\), it is important to analyze how the exponential function behaves. Since \(e^y\) can take on any positive value as \(y\) varies over the real numbers, \(\ln x\) covers all real numbers as its output values. Therefore, the range of \(\ln x\) is all real numbers: Range: \((-\infty, \infty)\)

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Most popular questions from this chapter

A glass has circular cross sections that taper (linearly) from a radius of \(5 \mathrm{cm}\) at the top of the glass to a radius of \(4 \mathrm{cm}\) at the bottom. The glass is \(15 \mathrm{cm}\) high and full of orange juice. How much work is required to drink all the juice through a straw if your mouth is \(5 \mathrm{cm}\) above the top of the glass? Assume the density of orange juice equals the density of water.

Show that the arc length of the catenary \(y=\cosh x\) over the interval \([0, a]\) is \(L=\sinh a\).

Use l'H么pital's Rule to evaluate the following limits. \(\lim _{x \rightarrow 0} \frac{\tanh ^{-1} x}{\tan (\pi x / 2)}\)

Suppose a force of \(30 \mathrm{N}\) is required to stretch and hold a spring \(0.2 \mathrm{m}\) from its equilibrium position. a. Assuming the spring obeys Hooke's law, find the spring constant \(k\) b. How much work is required to compress the spring \(0.4 \mathrm{m}\) from its equilibrium position? c. How much work is required to stretch the spring \(0.3 \mathrm{m}\) from its equilibrium position? d. How much additional work is required to stretch the spring \(0.2 \mathrm{m}\) if it has already been stretched \(0.2 \mathrm{m}\) from its equilibrium position?

A power line is attached at the same height to two utility poles that are separated by a distance of \(100 \mathrm{ft}\); the power line follows the curve \(f(x)=a \cosh (x / a) .\) Use the following steps to find the value of \(a\) that produces a sag of \(10 \mathrm{ft}\) midway between the poles. Use a coordinate system that places the poles at \(x=\pm 50\). a. Show that \(a\) satisfies the equation \(\cosh (50 / a)-1=10 / a\) b. Let \(t=10 / a,\) confirm that the equation in part (a) reduces to \(\cosh 5 t-1=t,\) and solve for \(t\) using a graphing utility. Report your answer accurate to two decimal places. c. Use your answer in part (b) to find \(a,\) and then compute the length of the power line.
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