Chapter 4: Q. 21 (page 399)

Use the new definition of $\ln x$ from Definition $4.36$ to argue that
$\ln x$ has domain $(0, \infty)$ and range $ℝ$ .
$e^{x}$ has domain $ℝ$ and range $(0, \infty)$ .

Short Answer

Expert verified

Part $(a)$ : $\ln x$ has domain $(0, \infty)$ and range $ℝ$ .

Part $(b)$ : $e^{x}$ has domain $ℝ$ and range $(0, \infty)$ .

Step by step solution

Part a Step 1. Given information

$\ln x$ has domain $(0, \infty)$ and range $ℝ$ .

Part a Step 2. Consider the following figure of ln x.

Part a Step 3. In the above figure, it is seen that the function ln x is defined on 0,∞ as it is continuous on 0,∞.

So, the domain of $\ln x$ is $(0, \infty)$ .

Now, $\ln x$ is role="math" localid="1648817423905" $0$ for $x = 1$ .It increases without bound as $x \to \infty^{+}$ and decreases without bound as $x \to 0^{+}$ .

So, the range of $\ln x$ is, $(- \infty, \infty)$ .

Therefore, $\ln x$ has domain $(0, \infty)$ and range $ℝ$ .

Part b Step 1. The objective is to argue that ex has domain ℝ and range 0,∞.

Now,

$e^{x} = \frac{1}{\ln x}$

Hence, the domain for $\ln x$ becomes the range for $e^{x}$ and the range of $\ln x$ becomes the domain for $e^{x}$ .

Therefore, $e^{x}$ has domain $ℝ$ and range $(0, \infty)$ .

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Use the new definition of $\ln x$ from Definition $4.36$ to argue that
$\ln x$ has domain $(0, \infty)$ and range $ℝ$ .
$e^{x}$ has domain $ℝ$ and range $(0, \infty)$ .

Short Answer

Step by step solution

Part a Step 1. Given information

Part a Step 2. Consider the following figure of ln x.

Part a Step 3. In the above figure, it is seen that the function ln x is defined on 0,∞ as it is continuous on 0,∞.

Part b Step 1. The objective is to argue that ex has domain ℝ and range 0,∞.

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Use the new definition of lnxfrom Definition 4.36to argue thatlnxhas domain 0,∞and rangeℝ.exhas domain ℝand range0,∞.

Short Answer

Step by step solution

Part a Step 1. Given information

Part a Step 2. Consider the following figure of ln x.

Part a Step 3. In the above figure, it is seen that the function ln x is defined on 0,∞ as it is continuous on 0,∞.

Part b Step 1. The objective is to argue that ex has domain ℝ and range 0,∞.

One App. One Place for Learning.

Most popular questions from this chapter

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Mechanics Maths

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Use the new definition of $\ln x$ from Definition $4.36$ to argue that
$\ln x$ has domain $(0, \infty)$ and range $ℝ$ .
$e^{x}$ has domain $ℝ$ and range $(0, \infty)$ .