Chapter 4: Q. 21 (page 399)

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

Short Answer

Expert verified

Part $(a)$ : $\ln x$ has domain $(0, 鈭�)$ and range $鈩�$ .

Part $(b)$ : $e^{x}$ has domain $鈩�$ and range $(0, 鈭�)$ .

Step by step solution

Part a Step 1. Given information

$\ln x$ has domain $(0, 鈭�)$ and range $鈩�$ .

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 $(0, 鈭�)$ .

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

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

Therefore, $\ln x$ has domain $(0, 鈭�)$ and range $鈩�$ .

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 $鈩�$ and range $(0, 鈭�)$ .

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91影视

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

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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91影视

Use the new definition of lnxfrom Definition 4.36to argue thatlnxhas domain 0,鈭�and range鈩�.exhas domain 鈩�and range0,鈭�.

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

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Discrete Mathematics

Calculus

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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